 So, this is the first lecture, and during the first lecture I wanted to focus on supersymmetry. Since my lectures will be dedicated to relation between supersymmetry and integrability, I have to review both parts, so today I will be mostly doing supersymmetry, and next lecture I will be probably mostly doing integrability, but there will be some integrability today also, and next lecture will be some supersymmetry. So I want to start with supersymmetric salaries with four supercharges, and I want to think about these salaries in two dimensions in one plus one dimension are called n equals 2 salaries. I want to think, for starting I want to think about two-dimensional space to be a cylinder, so this is a circle, and this will be time direction, but later there will be lots of discussion when this is arbitrary surface with the genus G and n punctures, and later even I will later, there will be three-dimensional, four-dimensional, five-dimensional, and maybe even maybe even six-dimensional salaries, about which I will think as two-dimensional salaries with infinitely many fields. Let me already here say a few words about what I mean by this, and for most of you, I actually designed these lectures to be very elementary, but I think that there are things that I will be discussing that I am not expert myself, but I still try to make it elementary. So the way you think about this is like if you are in three-dimensions, you can think about three-dimensional space to be this cylinder that I drawn here times the circle of some radius R, and that three-dimensional salary viewed in terms of the salary on this one will have infinitely many fields, which are usually called Kalutzakline modes, and they will be massive, and masses will be proportional to Kalutzakline mode divided by radius. If you are in four-dimensions, you can again think about four-dimensional salaries as times two torus, and again Kalutzakline modes will give you infinitely many fields in this two-dimensional salary on a cylinder, but there is an alternative way of thinking about the higher-dimensional salaries to be lower-dimensional, and this is achieved by omega background, and only think we will need from here, since I assume that we are familiar with Kalutzakline theories, is that the idea of Kalutzakline theories, this circle is related to the fact that we consider identification of the space coordinates along, let's take this R1, which we will make as we identify coordinate here, let's suppose to be X, we identify X and X plus R, and this is a shift, so we are using the transformation from translation group, and that leads us to compactification on a circle, and omega background, only think actually we have to think about this, and instead of such identification we take R2, let's say in the higher-dimensions instead of T2 we take R2, and we use the rotation with angle which we call epsilon, so this kind of thing I will call instead of Kalutzakline R2 epsilon, and it's actually a simple thing to see from the point of two-dimensional salary when we replace T2 by omega background with this thing, this will be again the salary with infinitely many fields, and masses will be related to the modes of the expansion in spherical function, so that's what I said here, is that we are studying now four supercharges in two-dimensions, but the salaries are not necessarily two-dimensional, they can be Kalutzakline from three-dimensions, four-dimensions, actually five-dimensions, or omega backgrounds coming from four-dimensions, this requires two-spaces real-dimensions as minimum ones, so we can think about omega backgrounds coming from four-dimensions or six-dimensions if we have two angles like epsilon, so this is a setup and now I want to say a few words about the algebra for supercharges. So first statement is that space of vacua, I need to explain this, which means supersymmetric vacua, in salaries for supercharges carries a representation of commutative associative algebra which is called, and we will be using that all the time, this algebra is called twisted chiral ring, so this is something that we are first interested in, let me introduce this object, in order to introduce it we have to remember that four supercharges means that we have, let's label them, q plus, q minus, q plus bar, and q minus bar, this is just notation of four charges, and we need only non-trivial anticommutation relation is q plus minus, anticommutator with q plus minus bar is two times Hamiltonian plus minus momentum, so this is superalgebra, rest of the stuff is trivial, and if we want to write Hamiltonian, if we want to cancel this plus minus translation, we can introduce two linear combinations, two independent linear combinations, the other ones will be related if we want to make with some coefficients, qa will be denoted by q plus plus q minus bar, and qb is q plus plus q minus, so with this notation you can check that Hamiltonian, yes, and the qa square equals zero, and qb square equals zero as a consequence of supersymmetric algebra, and you can check that Hamiltonian is harmonic in both, so we can write it as a qa bar or qb, qb bar, and I use just the word harmonic in quotation mark comparing the plus being dd dagger plus d dagger d and d square to zero, so it's just for analogy, so this is Hamiltonian and now the power of this supersymmetry with four supercharges is, by the way all these things I'm talking about goes back to Witten in late 80s and very well used than described by Czechotin and Waffa also in early 90s, Witten is probably late 80s and Czechotin and Waffa in early 90s, so there is a statement that A, if psi, so question we are asking is, since q square is to zero what is the cohomology of q, so I will pick one of them, let's pick qa and we now just denote it by q, so question is, what are cohomology classes of q, A or B doesn't matter, so I pick A, so first statement again is that suppose psi is such a class, which means that psi is a state in our quantum system, which is having the symmetry, any theory such that q psi equals zero and psi is not q of something, so if psi is an eigenfunction of Hamiltonian with nonzero eigenvalue or linear combination of source and lambda is not equal to zero, it's trivial to show that psi is actually trivial in cohomology, and where you show it you just write H psi as q q dagger plus q dagger q of psi, which is because we are talking about psi that is killed by q, so this term is zero, so it's q of q dagger psi because this term is zero, and we got a statement that we can write psi as q of one over lambda q dagger psi, which means that it's a q exact and it's trivial in cohomology, so for psi's that span the nonzero eigenfunctions of nonzero eigenvalues of Hamiltonian, this is trivial in cohomology, so the only non-trivial cohomology classes are corresponding to lambda equals zero, so we are now discussing, we are just saying that the vacua which preserves supersymmetry, supersymmetric vacua are maybe giving cohomology classes of q, so let's test this statement, so we assume now that H of psi equals zero, so now let's multiply it left by psi and calculate the norm, this is the place where I'm using the norm, normally in this discussion you don't have to use the norm, but to prove this statement I have to use the norm, so I assume that in Hilbert space of the vacua there is a scalar product, so this from the definition of H is nothing more but absolute value square of q psi moduli square plus q dagger psi moduli square because H is Laplace kind of operator and this is equal to zero from which I conclude assuming that the norm is non-degenerate, that q psi equals zero and q dagger psi equals zero, so which is stating that the psi which is a vacua are harmonic representatives of co-cohomology, now there is one more thing you have to do to prove that there is a one-to-one correspondence between supersymmetric vacua and co-cohomology is that if you assume that psi is trivial, so it's q of alpha, you have to end psi is the eigenfunction of Hamiltonian with zero eigenvalue, you have to deduce that psi is trivial or it's actually zero, the simple calculation shows that H psi is q of H alpha and is equal to zero, so if H alpha is not zero you conclude that alpha is q of something, which implies psi is zero, the very simple proof, now if you assume H alpha is zero it implies that q on alpha equals zero from which it follows again psi equals zero, so what I just explained is that there is really one-to-one correspondence between q-cohomology and the vacua and if we want to count supersymmetric vacua equivalently we can ask the question to describe co-homology classes of q, so this statement was independent of choice of this color product except the proving that it's a harmonic representative, so now another thing I want to use, go from states to operators, suppose OI is in q-cohomology as an operator, which means that the q with OI equals zero up to the shift of OI plus delta of some phi, then important thing and this is what I stated over there, the space of vacua is a representation of commutative associative algebra is that if O is on co-homologies and OI times OJ and this index I takes value somewhere, now we will actually we are studying where it takes the value, so OIOJ also will commute with q, it's a trivial calculation, so we can write that see, now let me put x here and the y, they can be in any position when x and the y are coordinates here and we forget this x and so on, not to mix with that thing, well the other thing with the important one is it's the OI as a function of x actually is independent of x in co-cohomology and this is consequent to the fact that if I take and shift x by delta x, so if I move the point then this will be OI of x plus q of something and the reason it's q of something is because shift operator which is p can be written as a q of something from this formula, so since x plus delta x is the same here as if I would, so this thing is the same as OI of x plus delta x times p acting on OI of x, p times O is q exact, it means that the OI of x is independent of position, this allows to take this formula for q commutator with O's and conclude that OI as now I drop the dependence on point times OJ can be written in terms of the some other O's which also have the properties that they commit with q, so now this indices run in the space of all operators which commits with q plus q of something and this statement is the one that I wrote in that side of blackboard, moreover it's symmetric in i replacement by j and this is because we are in because we are in two dimensions or higher, this will not be true in one dimension because ordering in two dimensions and higher makes no meaning since anywhere I put here x and y which I said is independent, I can now make x to move around y anywhere and I can't say what is the order because dependence on position of this guy whether it's first or the second is a dependence on x and this one I already proved that is trivial, q exact and since I'm living in q exact relations then this guy is symmetric in ij, so we have now symmetric matrix c i j k where indices i, j, k and everything I will be using belong to as they are labeled by the vacuums, let me explain this last statement and we can move on something interesting, suppose we found such one operator O i then it's obvious or let's say differently, suppose we found one vacuums which is supersymmetric, let's call it O, then we have to discuss uniqueness of these one states that we take and on this part actually I advise to read the old paper of Czechoslovakia from 1992, suppose we find one state special state in space of vacuums this is defined as all psi's annihilated by zero or equivalently q psi equals zero divided by psi identified with psi plus q of alpha, suppose we find such one state and now we have the operator O i which commits with q, we form another state which is O i acting on zero obviously this also belongs to vacuums because q will annihilate the i since q anticommutes with O, so now we found as many vacuum states as we have operators O, so there is a statement that there is an operator state correspondent which we will be using, there are some subtleties which might be related to the structure of this constant C i j k but when I will move to topological field theories I will discuss those subtleties but in the generic situation there is complete operator state correspondence and as I said the proving it is trivial and now we ask two type of questions, first type of question would be but you don't show that the structure is space or neither is reduced, what do you mean it's a reducible representation of the commutative ring or yeah it's a reducible, reducible, okay where we think we show that if there is operator O there is a state which is the zero eigen state of Hamiltonian, now we will start the structure of the vacuum state, all of the states, so this is the next step, so first we think we needed was this operator state correspondence and it sometimes fails but we are in generic situation, now we are discussing what kind of structures this space has, so there are two important things, first we know that it's a representation of the commutative ring, what kind of representation is we will be discussing during entire thing, this ring by the way let me give it a name if I did not give this ring is called twisted chiral ring, so what we have here, second that we have some basis in this state of vacuum described by this state's eye and it's very simple to show that operators O, let's call it O k's are not diagonal in this basis which means that the eye is not eigen vector of all operators O k, because if it would be eigen vectors then you get stuck, you start acting on it and you get stuck, you almost don't commute, almost commute, that's what I said I mean, or cij is symmetric, I said cij is symmetric you were here when I argued why it's symmetric, it's important, but I was meditating, oh what you are meditating about, more important than this and I would like to hear okay so this is a commutative ring and so we have a basis and we have this basis and as many commuting operators which is set of O's, now classical version of this quantum mechanical statement would be we have a phase space which is a replacement of the space of states and we have Poisson commuting operators and they're exactly as many as a half of the dimension of a phase space, so classical version of this is called classical integrable system, classical version is a phase space of some dimension let's say 2n and n commuting Poisson commuting and let me call now those h, a1 to n, so this would be a classical version, but we have a quantum version and this is quantum integrable system, so this statement that was there, the space of vacuum of supersymmetric supersymmetric theories in two dimensions with four supercharges carries representation of commutative associative algebra is the same statement as that the vacua and the operators O's are giving us quantum integrable system and this is generic actually and the space of vacua, let me one second so space of vacua now will be, I will write the last definition, is completely identified with the cohomology of the operator Q, yes that's what we will be doing in my lectures that's lectures are about this okay so there are two, one good news here and one not so good news the good news is that see we made very general statement that if you study the space of supersymmetric vacua you might learn something about quantum integrable systems but what quantum integrable system you have, I say nothing about it right, so we somehow have to identify what quantum integrable system it is, actually the space of supersymmetric vacua don't have to be, does it have to be finite dimensional, there are many simple examples you can give that this space of supersymmetric vacua can be infinite dimensional, not necessarily finite dimensional and then this adult starts, what do you mean by this, but when it's finite dimensional Hilbert space you're in a very good setup and you can ask the questions of what you just asked me so in old days I mentioned here paper by Czechoslovakia, what they were this studying in those old days was what's called a vacuum bundle, so let me give you some flavor what vacuum bundle was and I want to formulate the older questions and then show why newer questions are interesting and newer questions kind of started with the paper Greg Moore Nikita Nikrasov and I wrote in the mid 90s, a few years after the Czechoslovakia paper, so in old days what they would consider is a take space of parameters of your n equal to d equal to quantum field, sorry it comes with some number of parameters let's call this space of parameters local coordinates t and in any point here this is some point in the space of t there is a space of vacuums right because space of vacuums is in any quantum field theory in two dimensions it can be when you vary parameters the space can change so you vary the parameter and now think about this is a vector space and you have some parallel transport you are moving along the base so this entire total space they called the vacuum bundle which comes with some connection let's call it nabla in local coordinates this is I since I denoted this by ti and what Czechoslovakia and Vafa showed that this connection if deformed by when you take a state here you connect on that state as I said by oi and create another state in a vacuum so you can actually combine with arbitrary coefficient actually this connection that you had in this in the bundle plus some coefficient times the action of the operator oi this called this connection nabla twilder and they showed that it is flat so they would study in a sense many things about this flat connections in the space of vacuum bundles for example they could have studied some equations like this in the total space okay the flatness conditions they call t t star equation because it's flat as there is a conjugate one and the commutator of this which conjugate is zero and the conjugate one would be the differentiation in a complex conjugate parameters and so on and then there is entire business of studying the topological quantum field theories and the my lectures will not be about this although if there is some interest I can connect my lectures to to this stuff especially in December there was paper by uh nitzke cekoti and wafa or gayoto nitzke nitzke nitzke cekoti and wafa where in the language of the vacuum bundle and t t star equation they reinterpreted these things that see will be the main part of my lecture so in principle I might get back to this explaining how these guys relate this one stuff to another okay now one important question that everybody who works in quantum integrability asks there was this assumption that there is unique state in the space of vacuum from which we start to construct by acting the operators if we don't have this unique state we cannot really have operator state correspondence where I describe for example if this is not one state but million degenerate state and so on we are kind of in trouble in this identification but there the cekoti and wafa gave two arguments one argument was if this theory is super conformal if this theory comes from super conformal which means not only this algebra but the this is super Poincare algebra so you have to write super conformal algebra then there is something called u one current in n equals two super conformal algebra again this is not part of my talk but I need the argument in n equal to super conformal algebra there is a u one current and as such there is a spectral flow and spectral flow is a relation between navier-schwarz and ramon sector this theory is supersymmetric and there are these words for someone who did not study the subject that navier-schwarz and ramon means that you have fermions in this theory and if you have fermions you can ask questions what kind of boundary conditions you put I'm on a cylinder what kind of boundary condition you put on a fermions and you can put periodic or anti-periodic so if you put anti-periodic boundary conditions we are talking about navier-schwarz sector and if you put periodic boundary conditions you are talking about ramon sector these vacuum I'm describing are in ramon sector this vector zero is in ramon sector because I put periodic boundary conditions on fermions I hidden it somewhere here but if you look on the arguments it's there so if you want this state to be unique and if you are in a super conformal setup then it's extremely simple argument what you do you take navier-schwarz sector and it corresponds to anti-periodic boundary condition and there is unique vacuum there in navier-schwarz sector because of anti-periodic that there is a unique vacuum and then you have u one current which relates any state in navier-schwarz sector to any state in ramon sector by what's called spectral flow so in order to describe spectral flow you have to bosonize u one current and construct the operate spin operator and then spin operator allows you to connect any state actually it's smoothly connected every state in navier-schwarz sector is smoothly connected to the so if this is navier-schwarz sector this is its vacuum and this is ramon sector then there is a spectral flow from one to another and gives a unique state it's very important in super conformal algebra that o p is of a spin operator spin operator has unique o p is with any other operator this is important because you take this state and unique state gives a unique state if the salary is massive if it's not super conformal so this was about super conformal in super conformal there is unique state which leads to this zero in ramon sector and this is the state which comes from spectral flow from this minimum and this actual guy has its own way to describe is that if you have super conformal theory then you have left moving sector and you have a right moving sector and you have two u one charges which are the left moving one and the right moving one so every state in a vacuum in ramon sector will be labeled by two numbers now mathematically this is a statement about that see you can you are in a complex setup your vacu are described kind of in in in a in a language of the complex geometry and then you have p q forms it's not only that degree of the form you can count but you can count the holomorphic and anti-holomorphic part of the degree of the form and that's the relation that you can calculate so this guy then is a one which has the difference equals zero so this is a minimal one now if there is massive there is no you want current there is no super conformal algebra let me finish the argument but still there is identification between the usual arts and ramon sector and it goes like this you take disk you have a periodic anti-periodic boundary condition you put the spin operator it will make it periodic so spin operator always exists you can go from the periodic to anti-periodic boundary condition and this guy identifies every state here with here and interestingly it is actually true in n equal one theories that the kumrun wafa and i explained in 1994 for n equal one theories which are also related to the special geometry manifold so there is no spectral flow but there is kind of the spectral jumps that for every state in ramon sector there is a state in navier schwarz and other way around so we take this guy which is a minimal one ends around the argument then there is the correspondence yes we are going there now i'm going to write lagrangians and hamiltonians as i said did you come from the beginning of the lecture i said that it will be extremely primitive watch i will be doing extremely elementary there's two minutes okay so i said that it will be extremely elementary you don't need any knowledge except quantum mechanics okay some quantum fields or some kind of calculation uh uh knowledge of calculations of elementary things so anyway so the i argued now in quantum mechanical language in using fermions with periodic anti-periodic boundary condition that there is always unique state if there is a four supercharges and this unique state comes from the vacuum of navier schwarz sector so i'm using this to use the operator state correspondence now i'm in a business to ask the question why don't i take now these oh i these operators which form the commutative algebra and find common eigenstates of these operators so i'm asking the following question find common eigenstates of o's now obviously this statement is kind of wearing homology so this statement is uh i mean if i take functions of the o's these things they are commuting are also good commuting operators so i have to introduce some bases of the o's and then in that basis of the o's and i know how to introduce that i spent some time arguing on it suppose sigma labels the common eigenstates and the statement is that i will find the energy spectrum common eigenvalues and eigenstates of these o's which is different than the topological quantum field theory staff asks when it says take these guys you have the flat connections diagonalize c i j k okay because it's symmetric and so on then then calculate correlation functions of the operators o and write generating function of the correlation functions of o's which will be z of t will be correlation function like this and calculate so this would be question of topological quantum field theory i'm asking different question is to find the common eigenstates and why i'm asking these questions because this statement i made that why have a quantum integrable system this is a natural question in quantum integrable system to describe exact spectrum later we will find relations as i said and there is a paper i wrote within the cross of recently which had that so if there are no other questions i want to go to action functional space of fields Lagrangians what are the parameters to give some feeling what these theories are more questions and i do have a question yes so who tells you how to choose the o's since they all commute we could usual question in question then the i sigma would change right yeah but if i find one set i can find another set if you give me the change okay so i have to pick the set and i will be picking it in some clever way okay so if you wish as i i had here important things that these are two type of theories one which comes from kalutza Klein reductions and another which comes with this kind of thing now i want to emphasize important difference between these two these theories don't break Lorentz invariance so these theories are Lorentz invariant in higher dimension right these theories will be Lorentz invariant in lower dimension but the rotation with epsilon will break the Lorentz invariance in higher dimension because if you take this this piece of r2 your theory is a cylinder times this r2 with epsilon and the epsilon rotation broke the Lorentz invariant in in the theory so higher dimension theory will not be Lorentz invariant lower dimension theory will be Lorentz line so i will be creating the theories in lower dimension and in this particular case somehow cleverly the knowledge of some things in higher dimensional theory will will tell me how to pick the basis in a lower dimensional theory but this question of the what is the right choice of the basis and so on was probably covered by vasili's lectures when he was talking about the more complicated four dimensional theories no so you just postulated what y's are the operators that generates the carol ring postulate right well anyways since in the models here all the theories here have been discovered before us by people working on quantum integrable system and even sold by the people working quantum integrable system so we can use the knowledge from there the models here that integrable systems that we get here they're a little harder and less is known about them but we we have some choices anyway so i will erase and then start talking about the Lagrangians and so on and regretfully people ignore this piece of the supersymmetric Lagrangians and all these things in two dimensions and wrongfully because there is lots of things done by supersymmetric community that is very helpful and relevant Werner Nam actually had a nice comment that he said now he understood integrability better because if it's connected to supersymmetric vacua and he believes he understands supersymmetries now it looks like that he understands integrability but not everybody knows supersymmetry as much as Werner Nam so i have to know these were very general things any theory with four supercharges has these properties what we are interested are much more refined details okay Lagrangians yes please here it has some dimension as this fiber queues here are there is one queue always so the oh i why do you say locally the fibers are described by oh i this is the same thing i have to introduce connection which means that i have to describe how the state changes when i turn on the parameter that's the way i okay so this setups that i had here is more like what's called the tangent bundles okay so it's the base and the fiber are same dimension well that's okay if you have these oh i's in some number you can turn on certain number of parameters in this quantum field theory parameter space and for each oh i you can turn the one which doesn't break supersymmetry means that the space of the supersymmetric salaries is labeled by coefficients in front of linear term when you expand in oh i but you can see the translation reverence not no let me answer this again so if l is a Lagrangian then l plus ti oh i sum over i is also a supersymmetric Lagrangian these answers okay max what do you want oh no no this we will get i just explained i i just explained that you are using same operators like in a tangent but likely like in the gravity you're using same operators to deform the salary and they this operators one defines the fiber and they also deform the salary so the on the base we have the formation of supersymmetric theory and the fibers are the fox spaces or vacuum spaces in in each parameter and i'm studying dependence on this so it's like well that's one of the reasons for example why relation between what cekot and wafa we are doing and what we will be talking here is little bit uh kumrun likes to call the t t star equation like generalization of hitching equation but hitching equation has different dimension of base and and the fiber it's more like a vector bundle once it is our equation is more like a tangent bundle okay so let me uh start i actually prepare this lecture really for myself also it's almost follows the books that you get the negrassov and i were writing and it was lost on a drop box okay so into dimensions here i will introduce coordinates there in i'm in lorenzi and signature but it doesn't matter x plus x minus will be coordinates on a cylinder and then i have teta plus teta minus teta plus bar and teta minus bar are anti-commuting variables associated to supercharges and i will introduce shifted coordinates y plus minus which is x plus minus minus i teta plus minus teta plus minus bar and all fields in my quantum field theory will be functions of y these are called the chiral super fields uh there are special uh super fields which are uh so these are in a matter sector the h field and its super partners are described by vector multiplet which i explicitly write just now here in two dimensions actually not not big deal so vector multiplet is a super field where its lowest component is given by gauge field where in two dimensions so gauge field has two components locally zero and one time time direction and the space direction there is a this is normalization there is a complex color which i call sigma sigma which multiplies teta minus teta plus bar it's complex conjugate which multiplies teta plus teta minus bar then there is um there are fermions anti-commuting variables which i write like two i this formulas you had to actually we had to we nick it and i we had to look in lots of literature because they don't really spell out very well in books by now maybe these are explicit formulas that people derived when studies the extended supersymmetry in two dimensions and they are actually important in calculations so lambda plus and lambda minus are while spinners and then there is an auxiliary term sorry i have to write here auxiliary field which is called h and it multiplies the highest power of titas and this is auxiliary field now long ago as i said the chiral multiplets let me describe it here chiral super fields i will call x and they are given by something which is a function of y plus square root of two again fermions which are called size here they are functions of y y it's a combination so this combination is picked so that the covariant derivative will kill the field teta minus psi minus y and there is auxiliary field also for chirals they are called f so these are two types of fields gauge field is here and here are some scalar fields but as you see also already in a in vector multiplets there is a scalar and it's complex this is just artifact of the supersymmetry so these are while fermions but we have complex color so there is an equivalent way of describing this guy which is in terms of what's called twisted chiral multiplet which is actually a chiral multiplet like above one but a special one and this guy is related i call it sigma it's related to vector multiplet and these are super derivatives super space derivatives i don't have to define it there is this is a derivative with respect to teta plus and this is derivative that with teta minus but it's covariant because of a super space so claim is that this guy sigma has same information as v and when we write zelagrangian and this is it has nice expansion its lower component is given by scalar now i expand this thing and then i'm done with this long formula so now i will replace the vector multiplet that usually people work with with a twisted chiral multiplet and i need that so i want to show i i'm just showing some little tricks that we are using and they are necessary look at this one so what we have here and why twilda i have to define so there is another one which is why twilda why twilda plus minus is x plus minus minus plus i teta plus minus teta plus minus bar so this is this is a curvature of the gauge field there is no gauge field in this formula only curvature enters so this guy the sigma knows f mu nu is geometrically more correct to describe things in terms of sigma because the vector potential is too much extra information this guy is written in terms of the physical degrees of freedom h is an auxiliary field again so we have f we have h auxiliary field lambda and sigma and if you look at this you might ask okay what's the difference this auxiliary field is a two form f zero one is also two forms so why don't i just rename rename h to be shifted by f but it's not possible because f is d a and as d a there are consequences h is arbitrary to form f is a curvature of connection so d a plus a square so you cannot really but then i will play some tricks about this definition so for example they have monodromes and so on so it's important that this guy is here and now lagrangians what is there again because is it just the opposite sign of y what what's in front of the i theta it's a plus now you mean this is the definition okay sorry sorry it's a plus yeah right opposite sorry why in the vector field you have just one component of the a field where is the a zero plus and one okay you are correct i just forgot to write it so there is a conjugate term here which is theta plus theta plus bar times a zero plus a one thank you okay so now what is a Lagrangian most general Lagrangian of n equals two theory in two dimensions is constructed by first remembering that gauge fields which are in sigma should be in a joint representation so sigma is in a joint representation of gauge group now i introduce a gauge group i have a gauge group i take a joint representation of gauge group i call sigma it's a super field means that so it depends on it's a function on super space but it's a special function of super space which is of the type here and this is in a joint representation and matter fields are in some representation r i call them x some representation r which is direct sum of some multiplicity space times irreducible and r i's will be irreducible irreps of g so once i give the representation the gauge group i fix then i give representation there is still some freedom left because some lots of parts in Lagrangian are now fixed i am restricted i give my action is very much fixed now so where's the freedom i show where are the freedom and these places are important so action is an integral over two dimensional space which i called x which is dx plus dx minus there is something called the d term which is integration over all four titas i have four titas and i have four titas because i have four q's everything is a function on this super space this forms a cylinder and first one that i want to get in zilagrangian is f mu nu square i want to get these guys square because it's a young mill sewer so how do i get f mu nu square is that i integrate over four titas sigma times sigma bar i have two titas here i will get two when i multiply by conjugate all of them will be independent so i will have this times itself and there will be some other terms so four titas eat up square of this and correct square the bar will be theta minus theta plus bar and this term will have f mu nu square so here is f mu nu square plus super things good another term i can write trace is in a joint representation so let me make tracing up joint represent anything like this plus now let me pick up any function complex let's say something which later will become keller potential so let's call it keller potential k now it depends on z and z bar so instead of z i put x which is a matter field instead of z bar i put x bar and now since it's coupled with a gauge field i have to write how matter fields are coupled to gauge fields and the correct way is to multiply by exponential of v over two where v is a vector super multiply this one and you see i cannot write the term using only sigma because sigma contains f mu nu does it contains a but the v contains a so in order to write the interaction of the matter fields with a with gauge fields i have to go actually to vector multiply and this is this will be supersymmetric for any k now this is not all so this was written in terms of the four integrations of tita so i have to eat up in the baryazenian integral for titas but there are two other terms possible to write which includes integration over holomorphic part only in a superspace over only two titas and the way i introduce everything there are two possibilities and these two possibilities of course are connected to existence of q a and q b these two different twists that i i wrote there or also to mirror symmetry so plus integral over d2x and now two integrations over tita i will write tita plus tita minus so holomorphic coordinates i denoted here tita plus and tita minus anti holomorphic ones will be conjugated so there is such integration and i can integrate arbitrary function of x like this and it will be supersymmetric plus complex conjugate because this is a holomorphic function of x doesn't depend on x bar so this Lagrangian will be complex unless i add complex conjugate here and this is called an f term that was a d term this is called an f term and now you can probably realize that equally i could have chosen different combination of titas to be my holomorphic variables like tita plus and tita minus bar so i can keep this one everything else will be related so let's write the other integral d tita plus d tita minus bar and now we have freedom here to write any function holomorphic function of sigma and again plus complex conjugate and all the parameters that york frolic wanted from me are here arbitrary holomorphic function w of x arbitrary holomorphic function w tilde of sigma arbitrary keller potential k and the the first term for gauge will be fixed because we want to get f me new square where is the one of a g square in f me new square it seems well it's not f is all these guys are with arbitrary coefficients so if you wish with arbitrary coefficients yeah sorry i just wrote a structure there is nothing else you can write okay most general Lagrangian now we have to play with it and our job is to use the general knowledge we have and see how much we can run the Wilson normalization group and things like that how much we can say about a structure of this theory when we change the energy scale okay but what we know that if its supersymmetry is not broken it always will have that form only difference will be this function will change this function will change this function will change and the trace sigma sigma bar also will change with something else which is a function of it but we have this freedom of the functions and this is a big power allows us to say lots of things so let me introduce two other symmetries here and then we can make a break and then then i continue with the solution if i am too slow please tell me and i speed up if i am too fast also tell me your guy i'm too fast for you i'm too slow what do you think normal good okay so then i will introduce what's called the flavor group because this is a quantum field theory there are lots of global symmetries i described only local symmetry the g was a local local gauge symmetry now what kind of global symmetries we can have first of all obviously the global symmetry we can have is already on the blackboard because in a multiplicity space when we expand the arbitrary representation in terms of the irreducible representations we have entire space mi for each irreducible representation and we can have unitary group acting there which is can be a global group so maximal global group g global maximal is a product of umis right that's the flavor group we just think about x is as quarks right they come a lots of them they're there for each quark there are m of them mi if i is labeled the given quark there mi of them and we can have rotation group mixing them and this is unitary group umi this can be a maximal one but of course of course this is broken there is a subgroup of g maximal which is actually global group and it can be broken by this function this function doesn't have to be invariant under this rotation this is a holomorphic function so we have holomorphic function which should be invariant under the entire rotation not necessarily it just have to figure out so trick will be that then there are other representation we will start mixing the representation so we have to find real global symmetry group and this will be important another thing is that everybody had to ask me who have taken quantum field theory course because there is something important if g has u one factor i did not say that g was a simple group or a semi-simple group or something g age group can have some u one factors and if there are u one factors we can define magnetic charges we can take some question if you assume that x form vector space here but yeah does it look this form a manifold we have this group activity called some manifold and what it does mean okay this is a linear space okay let me explain so the axis will be linear coordinates right so the those non-trivial spaces here this called gauge linear sigma model so the those other things are really this is invariant under g so i have to divide space where x lives by the action of gauge group and i will get something complicated so for example you can assume it's from the beginning you can assume it from the beginning best way to describe it take a linear space and take the action of the gauge group on the linear set so you have a factor and you're living on the factors already those subclasses of all theories lead to very non-trivial statements and we will see that you can start saying that the x is the coordinate x is the coordinate in some calabi or whatever okay so let's then say this way these are the two-dimensional two-dimensional gauged linear sigma models so these are two-dimensional theories which are gauge theories okay and a hex branch so this sector where the gauge field is we will call a column branch so fields a sigma and lambda psi sorry no lambda we will call column branch and the everything where these guys live we call a hex branch so if this hex branch and column branch are interacting you can describe sorry by integrating out gauge field so you are now living in a factor of the space where x lives under the action of the gauge group or if the hex branch is massive you can integrate out this massive guys and have infinite expansion of something on a column branch both effective actions are impossible to calculate so we will have to figure out something clever way of dealing with that but i want you to introduce two important parameters which will play the role which will be part of the parameters on the base in the vacuum bundle and then we make a break so if gauge group g has a u1 factor for example if gauge group g is a u1 itself or product of u1s you can introduce as you know very well something called the theta term in gauge theory which is some theta times integral over your two dimensional surface trace of the curvature f okay topological term and if you have many u1 factors you can introduce many of them if a labels so you want factors 1 to r whatever r is but this is not good enough for us because the theory has to be supersymmetric with four supercharges and turns out that this has a natural supersymmetric generalization and is described by that this kind of terms are included here and let me single them out they are included here so what we do we introduce another background twisted superfield sigma twilder and sigma twilder this is simple way to write the term symbol twilder will be will start this guy will be theta plus i r and we introduce many of them as many as we have u1 factors so this is written instead of that term plus whatever and then there is a term in a twisted one particular term i call it w0 or wt let me call which is equal integral d2x dt to plus dt to minus of sigma a trace sigma a and i put here twilder now if way it's written that it said sigma a which is a a billion one will have this guy and then i take trace of this which gives me f and there will be bosonic term here which will look exactly like this but what i wrote is supersymmetric because i wrote in terms of superfields everything within terms superfields is supersymmetric so this will be called theta term or complexified theta term which is uh this has a name the r is called just to remember for future it's called a phyliopolis parameter so complexified theta term t is the sum of theta plus i r where r is phyliopolis parameter so now we introduce we singled out the theta term complexified supersymmetric theta term in w of sigma another question you can ask how do i make the guys massive can i make x is massive an answer is yes that term the mass term is also included here and it's written in this one let me single it out because it's a tricky thing that in n equal to supersymmetric theories that the masses are complex numbers so how can that be complex numbers it's very easy that the masses of fields x are complex numbers and not real numbers they are called twisted masses and they are possible to write for any global symmetry generator for any global symmetry generator you can write the mass for the field x and how does it look like you need a global symmetry to write these masses they are already written in the color potential there but i will single them out you take trace in representation r or following thing take x plus sum i this is a Hermitian conjugate of the super field x exponential of vi twilda i call it tensor identity in representation r i times x so what i did i had this representation where everything lived i took identity in the reducible component and i did some super fit will twilda in this component so if i have a global symmetry which is unitary group subgroup of umi for that one i introduce the field vi and vi as a vector multiplet itself can start with some coefficients mi twilda theta plus theta minus bar so the vector multiplet was starting is the lowest component was this and these numbers now are arbitrary okay but in order to write this i need the global group to be preserved for every global group i take exponential of vi here and i get a mass and these are complex masses they are called m twildas are called twisted masses and why i call it why people call it masses is it because when you expand it in the super field go to the regular refills this will be quadratic term in axis it will be x dagger times x with some coefficient so those will call masses so i showed that this is a general Lagrangian the general Lagrangian contains possibility to make x massive general Lagrangian contains possibility to have these theta terms and which counts the monopole charge the the first chain classes of our g g bundle if it has a u one factor it has first chain classes and everything is on the blackboard now when i come back i have to construct that quantum integrable system that i was saying that for any theory there is a quantum integrable system i have to describe that quantum integrable system and describe the bundle vacuum bundle which with parameters which are in w w twilda and k okay shall we have 10 minute break or something sorry i have a small question why can't you have a twist of chiral method as well we can but you said it's the most general Lagrangian the twisted multiple all multiplets are particular case of chiral so chiral multiplet is just a multiplet which says that it's a function of these variables why not all arbitrary ones but spaniel plus y and then i specialize there is a vector one and twisted one and so on what i say that if you take yesterday by the way i was asking question in a column normal if i say that sigma is d d bar of v and if i say that in v whatever is tending in the lower component i call the gauge field then sigma will be function of the gauge field right i don't want to have many gauge fields in the chiral sector so what i say that i single out those which are called gauge fields and rest of them i call the whatever and you can put on x any restriction you want you can put x to be a vector multiplet in principle but i'm not considering that it's uh no but i think x was a function of theta plus theta minus right and the first solution is the function of theta plus theta minus bar well you see what i call twisted chiral multiplet is this and this multiplet comes in a joint representation once because i want to have a one gauge field if i break this relation and i take sigma to be chiral then it's in it's in x already i allow those in x but i don't allow gauge fields to be in x again this relation makes sigma components expressed to in terms of v components in v i define the gauge it's any function of sigma is allowed i'm sorry i don't maybe we discuss it after but you can look there is a specific one which i called the sigma and in that one i am allowed to put any function any holomorph by the way mirror symmetry replaces x by sigma okay so now i want to continue in mentality of the wilson renormalization group and ask the question that if i make all matter fields to be massive this is called the theory with a mass gap so i consider theory with a mass gap all are are massive then what i can do i can in pass integral and this is now general argument which can be which which is made mathematically precise in a little bit of the time in a physical description i can integrate out all massive fields well massive fields came in the action with something like x dagger x so they have a gaussian term and they're massive so i integrate them out and what do i get i get this action back but this will disappear because i integrate out all of them this will change right this term was distinguished it had no derivatives this will change let's call it effective sorry what i am saying this will disappear i integrate out x this will change will have form very similar to what was written here and will have other higher powers and so on many derivatives so what we end up we end up with a Lagrangian of this one that needs to be calculated and this one it's the formation of this one so if we now yes our derivatives will be in here this will change this will not be like that there will be some Lagrangian which will be k of sigma sigma bar with some unknown function okay which has to be calculated so what i'm emphasizing here that this procedure of integrated out massive fields go with derivatives in space divided by mass expansion okay there and structural Lagrangian still will be integral over d2x d4 theta of something which depends on sigma sigma twilda sigma bar plus integral of d2x d theta plus d theta minus bar of something which i call w effective twilda of sigma because there are no other fields axis are gone what i'm saying that some magically some god calls you and tells you this is an exact integral whatever it will be it has to have this form one integral over d4 theta and one integral over d2 plus d2 minus bar this is a very important input supersymmetry and holomorphy are heavily used this term is holomorphic in sigma we assume that the calculation of quantum correction do not break the holomorphicity okay and second we assume that this term can be calculated and in a quadratic part of this term we have non-degenerate metric okay so this can be expanded as a quadratic part and so on and around it and and this this is all why this happens because i assumed mass gap if i assume mass gap all the non-trivial terms are here hidden in k twilda and w effective now statement mathematically justifiable provable statement or let's say theorem and some people would consider the theorem w effective twilda of sigma is exactly calculable and is given by one loop answer only by one loop only there are no higher loop corrections so one loop means determinant sir is this function only a concrete slightly algebra yeah the sigma is in a joint representation of the algebra sigma is a matrix in a joint representation polynomial or might be logarith can be anything i will now give the exact answer so i'm not saying i'm not saying that it's only calculable that i will calculate it okay these theories in two dimensions should be renormalizable that's all i am asking only normalizable one so what means that the as a quantum field theory wilson effective action should have a meaning then i'm asking question what it is so take quantum field theories while winsville effective action makes me and ask a question now so what kind of form it can have on any scale or even better in the completely low energy what kind of form and i'm saying if supersymmetry is not broken just look if supersymmetry is not broken only thing i'm left with is one field sigma which lives in a joint representation of gauge group so whatever i will write is written in terms of that field sigma and the normalized constants of this and that and all these kind of things right so general story told me that i can have term without derivatives which we call super potential twist it effective twist it super potential and we can have kinetic term and higher derivative terms so i collected kinetic term and all higher derivative terms which are i mean calculable in derivative expansion in effective field theory language and i say since these all things can happen i cannot calculate this so let's ignore this part for a second can i calculate this plus complex conjugate here of course an answer is and it has been discovered first in old days by diva kia dada salamon son and the collaborators and lusher and it is 1970s for cpn model so the cpn model would be i take n copies of complex plane here and c's right and i take the gauge group g to be u1 so i have gauge symmetry of u1 and i have the n this is c and these are n of them so then the hicks branch will be cpn the theory and they calculated w effective on a Coulomb branch not on a hicks branch when this procedure calls calculating exact action on a Coulomb branch exact answer on Coulomb branch and as i said it's not doable unless i make all our eyes there all our eyes massive if i make all our eyes massive then this what i said makes sense if i don't make those massive i can't integrate more over there will be points in a parameter space when something will go wrong and something will go wrong exactly in a point when the masses of some fields are integrated out actually happen to become zero so i was not allowed to integrate them out they are massless fields so this is a philosophy now one important thing is that the bunch of all this stuff can be ignored after i do topological twist because after i do topological twist and i consider topological theories things mathematically are well defined this is an index atia zinger index calculation nothing more in order to get w you just have to do some index calculation even you can go farther saying that see let's define w effective of sigma which is equal to some determinant and then we can argue whether this comes in a physical theory or not but this is calculable and i'm giving now answer for what is the w effective of sigma for general theory okay now please keep in mind buzzer me buzzer me i like i enjoy okay so in fact i don't really know what you mean by f you see this is a little bit of a mystery if f w f well i what i said here was that i had a pass integral let's think about the pass integral and and i want to do its perturbative expansion like in quantum field theory and i have some fields in pass integral which are massive okay in perturbation theory each term i can expand in one over mass okay when mass goes to infinity these terms will vanish i mean this so infinitely massive heavy guys will decouple so let's do it now every power of one over m will multiply derivative because otherwise i so it will be a derivative expansion okay and i say that there is a notion of a term without derivatives when the power goes to zero right that's term i can calculate okay that's what i mean by w effective the rest of this stuff i cannot calculate moreover i probably will never be able to calculate i introduce the scale i said that there are masses so the all x guys all the guys over there have a mass right neither of this mass is zero so i have for each of the guys contribute i can actually even watch contribution of the first quark which has mass m and if it's i take the lightest one then i will don't care about anything else but that quark and so now what is claimed here is it's ignore all this complicated when you ask these difficult questions you're going to direction of something not discovering something i would discover not asking those questions because all these guys that you're asking about are in this color potential in these terms this one is uniquely defined this one comes from the original w effective that i had over there plus corrections and i will now show you corrections and it will realize that what i will write you already know since 1969 okay but i will be utilizing it in in a question i'm asking okay so let me give you answer first so as i claim because of this high supersymmetry of n equals 2 all the higher loops will cancel and this is a good news right in this object in the holomorphic object and only one loop i have to calculate and now when i write the formula i will recognize it well there i will do one more thing sorry the sigma is leaving in a le algebra of gauge group right so in this le algebra is the composition of plus minus and diagonal component okay so let's take positive root of this le algebra so positive root would mean that i am sitting somewhere so this is my matrix of sigma i am sitting somewhere here i claim that this guy this sigma also is massive is massive with the mass m mu where mu is a primitive root related connected to this one so if this is in a position i and j so i here and j here there is a guy here i and j i i and whatever j j and i have to calculate mass will be sigma i minus sigma j the mass of this guy will depend on diagonal components of the same matrix so as long as sigma i does it equal sigma j away this point these guys are also massive and i can integrate them out also this is called abelianization same theorem is true about that it's only one loop exact moreover this part of calculation is extremely trivial and people have seen it so now i write w effective which leaves only on carton of gauge group and let's label it suppose our gauge group is un now i restrict to gauge group to be un just for simplicity so this is a function of the diagonal components of sigma and as a super field it's super partners but since this is holomorphic and the super field capital sigma is completely determined by its bosonic part which is sigma i will be using just word w effective of sigma and i'm giving answer for this formula and then super symmetry when i replace sigma by super field will give me everything so w effective of sigma equals sum over b 2 pi i tb trace sigma so this is in case if i had u1 factors in the gauge group in this particular case i have only one u1 factor so this is a one term in general if i have many other u1 factors there will be the term plus trace in representation r which is representation over there of sigma now sigma leaves the sorry masses will leave there sigma plus m tilde times log sigma plus m twilder minus one plus 2 pi i rho times sigma whereas color product is in the algebra and rho is a half sum of the positive roots half sum of positive roots so this is the final answer for any representation r and if you want to write in the representation you like you just plug in that representation so it's an answer now this calculation actually in bosonic four-dimensional theory was first made by colman-weinberg so this is a colman-weinberg computation of the potential in derivative expansion that's what i was saying that yorg knows that okay what happens with super symmetry into dimension that it's exact okay then in a supersymmetry this was discovered actually in four dimensions by veneciano and d ankelovich after the work of the divaikia data and so on but we don't i mean these historic parts are important i probably should be careful giving credits but definitely venecian colovich was after that divaikia lucer salamonson and these people so this is a final answer and we will be using it in future from now on now what are the vacuum after i calculate this can i make any statement about vacuum by the way very important this answer is true in any dimensions if i was in four dimensions on t2 on torus this trace r is an infinite sum now right because r is infinite dimensional representation this is written for any representation so for example if the seawaring two dimensions came from c dimension on a circle i have kalutza kline modes for any mode which means that the sigma has to be shifted by kalutza kline mod n and i have to have sum over there over n in that formula and that will make out of that formula classical dialog it because if i write sigma plus n log sigma plus n i get l2 okay so the if you i'm if i'm in a four dimension this trace will contain infinite repetition of the kalutza kline modes with weights and i will get some other answer so this formula has all the answers for the kalutza kline compactification for the omega background i have to do some work and that's why i'm giving lectures because otherwise it would be not interesting so i explained the meaning now conclusion after having so are you assuming the mass of the kalutza kline modes is kept you integrate out the other mass only uh no the it's a kalutza okay it's very good very good thank you i said that for every global group every kartan element of the global group i can introduce mass right i need global group to write the x square term so shift in a third dimension which is a kalutza kline is a global group it's just affine it's an affine shift so n divide by r mass of the kalutza kline mode is a charge is a twisted mass for the affine shifts so it's already included if i say that the kalutza kline modes are included here kalutza kline masses will be complex masses part of the complex masses for global group which is the affine one and i will integrate them out also because they have masses as long as n is not equal to zero if kalutza kline mode is equal to zero this is my theory into dimensions right so i integrate them out also and integrate them out would mean that i have to sum contributions of kalutza kline modes and how do i sum it so there will be a scalar component the mass you see i have here formal exist now this m twilda suddenly becomes some complex number plus n divide by r for every m there is its kalutza kline tower which is z m plus n divide by r and i have to sum this over n this is infinite sum which is summed up to the die logarithm so if i have four-dimensional theory on tutorials i have to do the same thing for that there will be double sum now n plus n prime will come n plus tau n prime and i have to double sum and i will get the formula so what i'm saying is that the as long as my four-dimensional theory is written as two-dimensional theory with infinitely many fields those infinitely many fields are packaged in this representation with some m's and i take this r to be arbitrary my formula over there gives the answer that's final answer of course when you remember that these m's now are so many and come together this becomes very horrendously long formula in four-dimensions on tutorials this is very long formula but but it's doable let me say now a few words what i can use it for because yorg wanted from me that i had to give meaning of what is this lagrangian is and this is very complicated i mean steven Weinberg in four-dimensions spends lots of time of writing the effective actions and pi mesons and so on and i said all this difficulty i just put aside i assume that the guy upstairs knows how to settle those things and i'm interested only in this term now i may claim knowing this term i know supersymmetric vacuum okay this defines a supersymmetric vacuum and now beginning of my talk if we go i have to introduce these operators i have to describe all these things so let's go there differ yes it has this m mu it seems it doesn't appear in your you have m mu associated with sigma i when sigma j it is already there i integrate these are the masses so i integrated sigma ij i integrated sigma ij when i is more than j the masses of these guys are sigma i minus sigma j and when i had this log formula there will be for those guys there will be instead of m here sigma i minus sigma j and then i have to sum over the root system and that's how i get this is a contribution i mean okay this two pi i rho of sigma coming from there the identity it's it sums up so identity is like this it's a simple identity i don't have it anyway just because it's a sum to zero sigma i minus sigma j when you sum over ij sum to zero the logs sit sink times log can sum up to the this thing i wrote there alpha the row times sigma so this is effective action of sigma ij's when i integrate out sigma ij's it's the answer final answer sorry and if sigma i goes to sigma j then it's when sigma i goes to sigma j thing goes bad now we first write the answer we'll write the vacuum and then take that place what happens when sigma i goes to sigma j and study exactly in that place our answer is wrong what happens now what nikita and i played this kind of game that we know something might go wrong there so let's write answer watch it and see what happens in vacuum and what happens that the the vacuum leak they start leaking like like in the shower you know and you can study all these details but let's not go there we call it leaking vacuum and then we discovered that in some paper of written called phases of n equal to theory he did not put twisted masses so he could have found in a large volume limit of kalabiyahu which is a very famous example the written in that paper described large volume limit of kalabiyahu as in terms of gauge linear sigma model in that place for quintic for example you have a global group suddenly coming up there is a symmetry it's an houses symmetry once global group comes out you can put these masses so the from kalabiyahu you can go to this massive theory and leak to some other branch which is not included in super conformal cell but in massive cell you can leak and this is connected to your to your question but i want to jump see where it's going right now the speed i'm going right now is like that i need probably two and a half hour lecture not three times but maybe ten times so let me let me it will be speed up okay okay so how do i calculate from this thing how do i count the vacuum so i'm starting now question of counting the cohomology of this operator q and what it has to do with the function that i just wrote w effective of sigma okay good question and as i said cpn model is a one sigma model with a cpn target is one example other example is a grass manian when i take the different representation then there is a cotangent bundle to grass manian there are many many homogeneous spaces i can write as a theory with a linear space divided by gauge symmetry group but i will give answers for all together so there are two approaches of describing vacuum knowing this one approach is physical so physical approach and topological approach now the topological quantum field theories were introduced long ago probably the one of the first one was albert schwarz's paper in the early 80s and then witten's paper which made the subject existing and somehow so my friend badim kablonowski likes to say that when witten introduced the notion of topological quantum field theory he did say that but the people did not fix in their minds so topological quantum field theory is a perfectly physical quantum field theory it just describes only vacuum sector of the physical theory so people say oh why should i be interested in topological quantum field theory uh if i'm a physicist and then you tell them you know but it's topological quantum field theory are not useless ones they are describing vacuum structure are you interested in vacuum structure or physical theory they say yes of course i'm interested in vacuum structure but why should i study topological useless theories and then you have to explain the topological useless theories actually captures the vacuum structure it's a long discussion but if we do the vacua physical or topological language both works because topological theory is the same as a vacuum so in a physical theory i remind you that we had this curvature of the gauge field f zero one in i am again in the language of uh minkowski space but now i jump into the arbitrary remand surface and i replace this by statement that the curvature over any two cycle trace f i called it a in a component a one over two pi i is quantized these are called magnetic charges so in path integral i am not only integrating over the gauge fields and in two dimensions there are no gauge degrees of freedom because gauge field uh has two degrees of freedom minus two which is because of symmetry it's a zero degree of freedom so basically in two dimension you left with nothing but such the fluxes so these are the only degrees of freedom now we had lots of other matter fields and so on we have to take into account but this is very important condition so in path integral we have a sum we have sum over m we have to remember this now this is a little bit difficult to work with path integral like that so what i will do and this was invented by many people probably but what i am using is from some old work of mine with Andrey Losev and Nikita Nekrasov is that let's promote f into independent two form now what that means that what we have to do we have to add to the Lagrangian following term sum i equals one to r where r is a number of these one factors n i integral of sigma of fi trace of fi now what happens with this is that if i am integrating see this sum over n now in path integral sum over n will create this will be the sum in other sum this is probably what's Poisson's resumption formula is of delta functions of trace integral of trace fi is equal integer right because this sum of exponents of n i times this and this and so on has non-zero contribution only if trace integral of trace of fi is an integer but putting trace of fi equals integer is exactly what i did here so these two things are equivalent either things that you restrict trace f to magnetic charge m and calculate the path integral or insert in path integral the sum over n's and this is a version of the Poisson resumption okay so i will insert it like here and after that i can forget about that condition because it's automatically is now what it does here when i calculate w effective it changes w effective by adding the extra term from there because f is in the same super multiplet as sigma sigma was the lowest component of the super multiplet f was the highest component of the super multiplet it means that see i have to add the linear term in my super potential which is 2 pi square root of minus 1 times sum from a equal 1 to r n i sigma i is n a sigma a see this is some simple thing moment i add that term there and i have to sum over n's now i know exact answer for super potential and i can assume that f now is not restricted to this condition and after that i can now minimize my super potential because now there is no restriction on fields before there was super potential with restriction on the field satisfying this now there is no restrictions on the field and just have to minimize super potential and that's what i will do this is a physical thing and now i give you answer what happens when you constructs out of super potential a potential so potential now comes after you eliminate auxiliary fields potential is labeled with n because i have a potential in every for every fixed n i have a potential and it is equal one half g i j bar minus 2 pi square root of minus 1 n i plus d w effective twilder d sigma i times 2 pi square root of minus 1 n i bar n j bar plus d w effective bar this is from complex conjugate d sigma bar j and sum over i and j where i don't know what is g i j bar this guy i would know only if i would calculate kinetic energy here but i have not calculated i don't know what it is but i assume that there is some and then there is an answer from which i conclude that in every super selection sector given by this set of n's which is here the vacuum equation is d equals 0 because this is the absolute value square of some holomorphic function actually this is even clear this term here if i would include in w effective if i would write this new w effective then what's written there is new w effective prime moduli square right and this has a minimum when it's 0 and that zero condition now if i remember this the w prime is not the same as w this condition now is 1 over 2 pi i 1 over 2 pi square root of minus 1 d w twilder effective d sigma i equals n i so for every integer n an integer set so i take carton of gauge group i put in the diagonal in carton of gauge group integer numbers i order them i have ordered set of integer number i have this element and for each element like that there is a vacuum such that sigma solves this equation so now wakua are enumerated by number of this set of integers and the sigmas that i used to describe the effective salary solve the equation sigma is in a joint representation trace here in r is about these m's here okay so what do you have to do let me give example let me give example this is a mathematical way of writing trace in some representation if you know something in a joint representation how to rewrite the matrix from the joint representation to any representation for example if you have a matrix in a fundamental representation spin one half right what is the spin three representation you have to take product one half times one half times it's a symmetric power six symmetric power or whatever or spin one half representation right you calculate that and then now we are in that representation and you take trace in that representation so you have to okay this sigma lives in a joint representation these guys masses live not even in the representation right they live somewhere else in the global group so this is a direct sum of a joint representation plus something written in representation r and then taking trace so when i will calculate with light express formula you will see it so that's technical this is a just simple way of writing the entire representation theory is about that so if you have spin one half how to construct spin five you just multiply many times construct spin five and then take a trace okay i will give examples you say sigma is somewhere in an actual representation but m is somewhere else but what do you mean by the sum of the two let me write this stupid thing for the un okay let me write for you and this thing this is some no it's it's a correct language that's how they write it sum over i and j sigma i minus sigma j times log of sigma i minus sigma j plus m minus one and i not equal j that's what is written there so let's stop to discuss okay okay i want to rewrite now this equation in a simpler form to just give a variant way of writing everything and then i move to topological language so right yeah for every n i have this vacuum and my field theory has a disconnected target so n gives me super selection sector for in supersets like that so the answer is sum over n and i will write those things now just stay with me this i fixed n okay and i described vacuum in that then i have to take all sets of n's and for each n as i wrote here you see the disconnected target space in each set there is a potential i minimize potential for fixed n and then i have to vary the n that's what it is but now i will eliminate n don't worry okay so let's take this formula and take exponent of the both sides i get exponential of dw effective this sigma i equals one so i eliminated the n all the n's are hidden all solution of this are good so i multiply two pi i take exponent and this has all va this describes all vacuum so i will when i will do the examples it will it will be little happier i hope okay now i derive this equation and i want to move the topological language which is extremely extremely valuable actually what is the topological field theory language for this business this how it was actually discovered first the first example was discovered in the paper of murne grassov and myself in mid 90s which was example which later turned out to be no linear scheddinger equation so we discovered the quantum field theory supersymmetric quantum field theory for which the vacuum sector was given by no linear scheddinger equation so this was uh first one and then we for some reason stopped thinking about that and then 10 years later gerasimov and myself got back to the subject and it developed and work with negrasov when we started almost everything possible and then with wasili best one we started even more things which are impossible and so uh let me give a topological language so now in topological language what will i will do is like this i had this sigma which was super field and it had gauge field fermion actually psi psi minus and sigma and sigma bar okay these were the fields in super field sigma so i topological theory changes the way i think about those guys so psi psi plus psi minus will be replaced by lambda which is now one form they were spinners lambda which is a one form and eta and chi which are scholars if you count how many of these guys are they're exactly as many as i just wrote because one form in two dimensions has two components and there are two others the four components and this was four components two for psi plus and two for psi minus this is the same thing now what the way it's done is that you take r symmetry r symmetry i don't want now to explain what r symmetry is but there is a u1 for r symmetry and change in a dirac operator spin connection which couples it to the metric and tells you what are the spins of these guys change it by adding the current the connection from that r symmetry u1 and that will change the things and the claim is that for a twisted super field sigma it's actually easy the same guys you just relabel them in terms of representation sorry just this is a one form instead of psi plus minus eta and chi these guys taste the same it is a scholar okay that's what you achieve now what about the twist so this is a regular twist for the two super multiplets sigma what about chiral multiplets what about matter sectors and here as a lots of thing happens now we have huge ambiguity because as I told you I used r symmetry u1 to change the spins of these guys and transform it to eliminate spinners and I have only one forms and scholars but once I go to the matter sector my global group has bunch of u1s right I mean this global group is a product of UMI there are lots of u1s I can take any subgroup of u1 and change the spin connection coupling to matter fields so when I write cover and derivative acting on matter field I will twist using the u1 from from the global group and there are lots of possibilities so twist in a matter center is ambiguous they're in two dimensions there are many many twists okay so what is the principle that will allow to have after the twist one of the supercharges q to be scalar because I want that after I twist the theory the one of supercharges to be scalar the one that squares to zero this is the one I called a before and turns out that this principle is following anything you do is allowed anything is allowed anything I mean any twist of the spin connection plus the u1 coming from any combination of u1s coming from global group is allowed with any coefficients here as long as w of x which is very important there is a w of x in the original theory becomes after you do this one zero form on a watch on sigma so any twist which will lead to one xenophore is good I will give you an example just just to make sure that we understand something and this is a main example that we will be working from now on so how do I change spin of the fields how do I do the twist of the spin connection there's this linear combination to have w of x becoming one form which is that it's a wz it acquires a worksheet index z okay how do I do that now obviously what I am talking about if my background metric in two dimensions is flat what I am doing I'm doing nothing I'm not changing anything I'm just changing its coupling to the background metric okay example I call it main example it's not the first example we discovered but this is a main example so take representation r to be l fundamental representations of grade group g which I take to be un take l anti-fundamental representations and one adjoint so this is my set of representations now I give the masses I call it m af a goes from one to l I call this one m af bar a goes one to l and I call this m adjoint okay they are all massive and w of x I will take such that yeah so global group maximal global group here g global maximal is ul times ul times u1 so you would think that I can turn on super potential which preserves maximal carton subgroup of this which is u1 to the power 2l plus 1 obviously generic super potential will not preserve that so let me give a super potential which is sum over ab and s of m ab s l twilda let me call this guy l twilda as a field this is l then they have indices because there are these indices a and a here so this I write l twilda a phi to the power 2 s a times lb now what this is phi is in adjoint phi in any even power any integer power will be also in adjoint so s a's then have to be half integers and then I have fundamental and anti-fundamental so this is trace is this clear I have fundamental adjoint anti-fundamental so this is invariant under the global group and what is a global group now I claim that this is invariant under u1 to the power 2l plus 1 only if masses are related in general it will not be it will break completely as group group what do I erase now I have nothing okay let me okay we forget now about all this stuff we don't need this anymore this representation thing so uh claim if m af is equal minus mu a plus i s a u m af bar equals mu a plus i s a u and m adjoint equals minus i u then following is a symmetry l a goes to exponential of minus mu a plus i s a u l a l a twilda goes to the conjugate and phi goes to exponential of minus i u phi and now you take these guys these transformations of course we have here u1 to the power l we have u1 to the power l here which is different and then we have extra u1 here so we get u1 to the power 2l plus 1 and for those masses this is invariant okay so now we have the global group which is huge u1 to the power 2l plus 1 and we can use the that to twist this error so claim another one is that in this one we can for example do what's called symplectic twist as I said there are infinitely many twists you can do and I will give a general one and this particular example I need for future so I keep it on the blackboard so symplectic twist is like this phi this phi over there this adjoint guy becomes one zero form under to a symplectic twist it's a one zero form so we call it phi z l and l twilda become one half minus s comma zero forms and since s is half integer it's one half times an integer then this is integer number and that's what it is and then w of x we can calculate now what it is so we have one half minus this is one half minus s comma zero form this will be the form of the type two s zero form and this will be one half minus s zero form so what happens one half plus one half is one minus s minus s is two s cancel this one so this is a one zero form now in general general if we use them in this particular example more most general twists then this guy is becoming u zero form for any u these guys will become one half minus s times u plus minus t comma zero differentials and that total thing will be one zero form again and this is true for any u and p so in two dimensions in this particular example there are two parameter family of twists you can do such that w will be um one zero four and now fun starts okay so fun starts like this once you twist the theory your Lagrangian after topological twist has many many interesting properties one important property is that the entire past integral entire past integral is an integral is integral only over what's called bps states bps solutions and now let me say what are bps solutions in most generality for symplectic twist they said there are other twists but i will not do that so bps equations are basically what it says that the entire past integral will have action upstairs will be key of something and i can drop everything which is key of something and what will remain will be basically the delta function of the equations i will write now so what will remain Fourier exponent of some Lagrange multiplier times equation and i will write this equation so entire integral becomes integral over solutions of some pds these are pds now what are these pds in a symplectic twist for this particular case or actually probably no most general actually most general i will now know most general because it can be written in most generals and will specify so we have dz bar which is a derivative with respect to connection which i have because i keeps the connection it was in super twisted superfield times xi where xi are my fields in representation are equals minus gi i bar dw complex conjugate of x complex w was a holomorphic so this is complex conjugate of x bar over dx i bar of bar this is one equation and another equation is f dz bar of this connection a so curvature of connection a plus mu of x comma x bar equals zero where mu mu is a moment map g star valued associated with symplectic form omega which is some ij bar gi j bar of x x bar dx i wedge dx j bar so i have this scalar metric gi j bar which enters in the equation and defines the g star valued moment map mu with the symplectic form calculated from the scalar metric entering in the one one component of the equation so this now system is invariant under action of the gauge group so i have to divide by action of the gauge group and usually this is usual in a good situation this is finite dimensional this modular space is finite dimensional and i have finite dimensional integral so i started with infinite dimensional integral i ended up with a finite dimensional integral why because if i'm interested only the correlation functions and this is what topological fields or is in terms correlation function so if q exact operators which i called chiral ring generators then only i care is non trivial terms in Lagrangian which are not q exact and all the terms are q exact i throw it away and what is remaining is only only the integral over the equations solutions of this equation and this is very powerful now because some many interesting equations have this form okay let's go to our main example and see what are the equations in our main example even if we will take and drop these guys and keep only a joint just let's take our main example over there make capital l to be zero and i have only one field phi a joint that can't be more simpler than that what is this equation for that case the answer is this equation in that case has a name and it's called the hitching equation let's write what it is there is no w star because i throw away i took the l equals zero i i i throw away this there is no super potential w so this guy zero and what is this one it's obvious what it is this moment map for that thing is a commutator of phi z with phi z bar that generates the algebra valued moment map with a symplectic form d phi wedge d phi bar so i get equation nabla x the x is phi now nabla phi equals zero obviously complex conjugate equations that's the phi has an index z nabla z phi z bar also is zero and fz z bar plus commutator of phi equals zero let me write it somewhere let me write it here so in the case of l equals zero the modular space i have is a modular space of solutions to the equation nabla z bar a phi z equals zero these are two complex equations and another one fz z bar plus commutator of phi z with phi z bar equals zero divide by action of the gauge group and this is called the hitching modular space this is hyperkeler manifold and lots of things interesting so by doing what we are doing by just stupidly calculating this function we are calculating cohomology of the modular space to solutions of the hitching equation we're counting for example we are calculating volume of the manifolds which is solution of these equations now these equations are written for arbitrary Riemann surface sigma so let's now replace our cylinder as i said by arbitrary Riemann surface of genus g and and marked points i had cylinder all the ways but as a filter i can put it in any manifold okay calculating intersection theory of of this modular space has been a very old problem hitching row this equation 1987 or something and this is a reduction of the instant on self-dual young mills equations in four dimensions and so on and now you probably realize that if i can do this i can do the intersection theory on modular space of instantons blah blah blah so we are now in the right setups but general equations are this so this is true for any w star for it's it's most general and nikita and i recently actually even wrote a paper of explicit answer for this story okay what does it mean and i finished today's lecture by this what does it mean that what i just said i wrote with nikita some whatever thing that means and the one application of the study of this modular of the equations is following there has been a standard law very standard since i was a baby that's the well since i was whatever very young men that the topological theory has a follow in supersymmetric theories are following identities take trace of minus one to the fermion number of exponential of minus beta h minus t i o i sum over i where o i are chiral ring operators now the argument standard lower gauss like this let's call this z of t well because of for every boson for energy more than zero there is a fermion these things will contribute with opposite signs and they will cancel actually here so this reduces only to the trace in the vacuum of physical theory of minus one to the f in the vacuum h has zero eigenvalue of exponential of minus t i o i okay little simplification we got but then there was an amygical step which is saying that this theory is some finite dimensional quantum mechanics the logical field theory is some finite dimensional quantum mechanics what is that finite dimensional space that finite dimensional space is this space which is finite dimensional and then they used to write that this is same as trace of exponential of minus t i h i in some quantum mechanics where h i's are associated to o i's somehow and these quantum mechanics we have to invent so now i'm giving answer explicitly in this language that this actually sometimes is true and until the some work that whatever i mentioned greg nikita and myself in mid mid nineties only example was known for such things the real real clean example was example of two-dimensional young mills which is empty salary but leads to many interesting stuff and so on so i claim now that this identity can be clearly written in the case of this okay and give you the answer which is from the paper i wrote within a class of about six months ago of the general case just quickly jump to the answer type of answer which will be the kind of announcement of the next week's lecture start of next week's lectures is at zeta apological for any theory of this type is equal sum over certain space b of h to the power g minus one of sigma b exponential of minus sum over i t i of o i evaluated on sigma b where b is a space of solutions of exponential of dw effective d sigma i equals one this is our equations that we derived for vacuum and h it has a name it is called the handle gluing operator and is equal exponential of minus certain function which has a meaning of the same as super potential but in two-dimension it couples to the metric so this appears only when you have a metric when you have a card space in two dimensions for cylinder is not there times determinant ij of two derivatives of our super potential times one Dermond of sigma where delta of sigma is a product of positive roots of alpha comma sigma which is in the un an case is just sigma i minus sigma j and that's it so that's an exact answer of for intersection theory you plug in in any function symmetric function on the carton of your gauge group solve this equation for w effective i had their formula for general case and take h to the power g minus one and some so this is a sum over all genes so i finished today and next week i start with topological theory and introduce algebraic integrable systems i will consider omega background and calculate same things for omega background with one epsilon so hopefully next week we will discuss some thermodynamic patterns thank you