 Okay, so it's 2.30, so we'll start and let me tell you something, okay, so in this course I'll tell you something about a circle of problems that is actually studied by some, as a part of quantum field theories and string theories and that circle of problems encompasses the following topics. We are looking at what we call supersymmetric field theories, gauge theories. It's supersymmetry in various number of dimensions. On the one hand, we are trying to understand exact correlation functions in the theories. These exact correlation functions are in quantum theories and by exact I mean that not only we are trying to sum up effectively all Feynman diagrams, but also to take into account the non-partibutive corrections and sum over all of these non-partibutive corrections and in some situations for certain observables that's possible and answers can be computed exactly. On the one hand, they display a rich mathematical structure and connect to other topics, so one particular connection that we'll be doing in this course of lectures will be to integrable systems, so that's completely integrable systems of classical mechanics and quantization of these systems. Okay, so since I was asked to give this series of lectures for non-experts, let me give a broader picture first and give an introduction to supersymmetry. So for that purpose I want to have some feedback from the audience to understand what I'm saying is well known to everyone or what's not. So let me ask a question. So if I write that list of dimensions, so real dimensions of a manifold M, what would be special about this list of numbers? D equals 3, D equals 4, D equals 6, D equals 10. Do many people recognize it as a special list of dimensions? Not really. Okay, very good. So then my first lecture is not useless and we'll talk about what's special for real smooth manifolds of these dimensions. Okay, so to start with let's consider a vector space or affine space, affine space V and symmetric B linear form on it G. So for simplicity first you can think about V to be just a complex space and G is a symmetric B linear form on that complex space. Now, what's that? The field is real complex. So for start you can think about field to be complex numbers then we can restrict real numbers. Okay, just analyze this is a bit simpler for supersymmetry of our complex numbers. So well first of all let's recall what is Poincare, what is Poincare group or the spare VG, well just isometry group and we'll call it iso. So it's just an extension of SOV of orthogonal group defined by the metric by the translations of V. So that's a semi-direct product with orthogonal group by the translation. Now the idea of the supersymmetry is to generalize the structure of the Poincare group to the super spaces or super manifolds. So for that one introduces vector spaces and does think vector spaces of non-commuting statistics namely that the coordinates in those spaces are anti-commuting and we'll call them fermionic or grassman and if you have some vector space U where the coordinates are usual bisonic so that X commute will denote Po the space with the same structure but fermionic coordinates so they so they anti-commute and call the p parity change and parity means statistics in physical language. So let me define the super Poincare super Poincare group well there's let's do at the level of algebra namely super Poincare le algebra so what is a super le algebra well it's just Z2 graded le algebra and that means that it splits into the bisonic part and fermionic part which I called here g0 and g1 that's degree 0 degree 1 and so that the elements of g0 are computing elements elements of g1 are anti-commuting elements I mean before we impose the structure of the le bracket and that means that the le bracket the le bracket from g yes to g so it's supposed to be so if you count the statistics here so it it it goes from 0 0 to 0 from 1 1 to to to to 0 or and from 1 0 to 1 so so then g1 is just is g0 module so if you fix the bisonic subalgebra of the super le algebra then g0 has to be the the the usual le algebra and g1 has to be a representation of that usual bisonic le algebra that's that's one thing and another one from here we recover that to have a le bracket defined on the odd elements so also I called sometimes I called these elements even and these elements odd yes so to have le bracket defined on these odd elements since they are anti-commuting then the definition of le bracket implies that this map from the grade 1 1 to the grade 0 has to be symmetric so the le bracket restricted to g1 is a symmetric map g1 times g1 to g0 symmetric okay so this is just general general definitions about super le algebras now let's apply that to the superponkare to the construction of superponkare le algebra okay so let's see what would we want to take for the g1 for the extension of the ponkare le algebra well we want to consider spinors spinor space s and for now s would be just is a clefert clefert algebra module for the clefeltal for the clefeltal algebra of the vector space v and the metric gene and the clefeltal algebra is what physicists call the algebra of gamma matrices so you take just a free tensor algebra of v and you you you quotient it and you quotient it by the relation that v v to 1 to v times v is the bilinear form computed on that element v times identity so it's well known that if you take the degree 2 subspace in that clefeltal algebra then the generators of it like anti-symmetric anti-symmetric elements of the of the square of v they act on the spinors they act on the representation of the clefeltal algebra in the same sense as spin representation of s of v or more accurately one should put it here spin v which is double cover of the s of v okay so now let's take the vector space to be v plus s and that what we'll call that's what we'll call super ponkare this symbolically it can be summarized that this is isometry group of this is super space which is an extension of the ordinary vector space v by the space of spinors okay now so what about Leibreck is here well the Leibreck to define the Leibreck let's look at this structure so we have the the isometry group of this space this is following first of all you have isometry group of v that's the usual ponkare so that's ponkare and then it extends by the generators in that spinor space so the spinor space is naturally a module of the ponkare algebra we know how to act on spinors by rotations of v and the trans I mean by Isov and the translations act by zero so to finish the definition of the of this algebra we just need to define this Leibreck in the degree one and that has to be symmetric pairing so we have to introduce so to give a Leibreck here in degree one we need to introduce a symmetric pairing namely symmetric map yes from s times s to v okay and that would be the odd odd bracket in the superponkare algebra isometry of f plus s so what's special about d equals 3 4 6 10 where d is dimension of v well namely in these dimensions there is exist a representation the d exist representation of a superponkare algebra I saw of v plus s namely there exists a suitable spinor module as slanted such that well that representation is called which is called minimal vector multiplet and vector multiplet means that the bisonic components of this multiplet transform as vectors and in addition there are super partners which obtained by action of the odd generators of the superponkare algebra on the on the vectors are fermions called the super partners of fermionic partners even part of the presentation yes even part of representations of vectors and odd part of representations are fermionic spinors of uh uh muscles yes yes yes thank you yes so no more muscles and fermionic partners are spinners in s so mathematically this is captured by the following observation well there is a theory it goes to bring schwarz shirk 77 and it tells the following well there is a necessary condition for existence of that representation which is summarized by this three-spinner three-spinner rule how it's usually called in papers and is the following consider this bracket let me denote it by angular parenthesis from s times s to v which is just the lea bracket on the superponkare algebra but it's odd i mean it's symmetric bracket because s is an odd subspace okay so and uh the three-spinner rule means that this equation holds true the sum over cyclic permutations of the bracket between two spinors psi one psi two which are elements here so that gives a vector and then you can act by that vector on the third spinner psi three where the this action means cleft-artal algebra action so cleft-artal algebra action means that well if you in terms of gamma matrices notations of physical literature if you have a vector v mu contracted with the gamma matrices gamma mu intact by that on the spine okay so uh this the three-spinner rule is the condition on this on this odd bracket which is that the sum over cyclic permutations of such three-spinner spinors and the theorem is it's not Jacobi identity no uh no no it's not because here we have a vector and we act by that vector by the cleft-artal algebra action Jacobi identity is automatically satisfied yes yes yes no it's not the Jacobi energy just look here we have a cleft-artal algebra action by by a vector on the spinner and it's not the action in the algebra in the in the supraprankary algebra on spinor sects rotation group but not translation translation doesn't act so it's not Jacobi identity and the the non-trivial observation of a physicist from 77 is that only only for dimension of v three four six ten exist the spinner module of v such that this holds no i think no no i'm slightly confused what was the definition of this symmetric map of this symmetric map well i i i haven't uh defined it yet i i i said that uh it can be constructed and the theorem is that in those dimensions it can be constructed and moreover it satisfies that condition and let me actually answer the question in this question since it's important i was going to answer it the definition of this symmetric map exactly exactly exactly so now uh now let's discuss what is s actually okay so first of all again let's work over complex field and let's v to be c to d and recall what is cleft-artal algebra of v uh well so it can bear okay in terms of generators let's say it's generated by gamma mu which satisfy the anti commutation relations so just uh just encompassing the definition of cleft-artal algebra of that abstract which is written in the upper corner of that legboard now uh it it's well known that it has a representation okay if so if d is even so it's represented on the space of spinors which can be decomposed into the spinors of positive and negative corrality and it's represented okay and s is c to d over 2 so so gamma matrices are 2 to d over 2 by 2 to d over 2 complex matrices that satisfies that commutation relation so uh and s plus and s minus are reducible so this is so this guy's irreducible spin v modules so if you like uh dinking the algebra notations then if d is even we are talking about so d yes and and those s plus and s minus modules are the modules that have the dinking weights here in in this node or in this node i mean where you you put one here or here and zeroes everywhere else you assume that d is even if d is even exactly if d is even then uh we have a positive and negative corrality spinors and that's what they are and in terms of uh in terms of gamma matrices or in terms of cleft-artal algebra there is a projection operation well uh projector again not literally projector but let's say corrality operator corrality operator it's an element of the cleft-artal algebra that and it commutes with all uh gamma mu and it's just gamma one gamma two and so on gamma d yes and uh the s plus and s minus are plus minus one against spaces of this operator it's gamma well yes yes uh since we are not necessarily in four dimensions that's called gamma star which is a okay uh if d is odd then the representation the spin representation doesn't split onto positive and negative corrality yes and uh you have just a irreducible representation of spinors then s itself so if if d is odd then s is c2 to t or d minus one over two and it's irreducible irreducible spin v modulus uh okay now uh c to power c b would be but two power is not here but two is two two power b oh yes yes yes thank you yes two to t to d over two thank you uh okay so we have a cleft-artal algebra module now on s actually let me uh fix notations a little bit or maybe simplified let me s without slant define uh to denote just the the minimal spin or representation i'm sorry i will remove this slant and i will use uh different font so just uh that s is what's defined here and the slanted s like here it will be the representation which is necessary to construct the massless vector the massless minimal vector multiplet and this s with this slant exists only in three four six ten ten and we'll write it in terms of the standard spinors of cleft-artal algebra in a second yes there are modules of the even part of the cleft-artal algebra or the spin inside the cleft-artal algebra i'm sorry i didn't hear quite question s s is a module of cleft-artal algebra okay but for example in the even case you you made the split yes and those are uh irreducible representations of the even part of the cleft-artal oh sure sure sure sure they are but but they are irreducible with respect to the the the complete one because uh absolutely because if you take uh the elements of odd degree here like like code number of gamma matrices and you add them by them on s plus they map it to s minus okay but then this is spin group inside the even part of the cleft-artal algebra yes is the uh real you mean you you want to talk about the real spin group uh so so far i i don't want to talk about reality conditions so far just let me work over complex field it's it's easier to analyze over complex fields uh so everything over complex numbers uh are the questions of this okay okay uh okay so let let let me write the answer immediately here to keep even mind and continue so as for that list of d equals three four six and ten the slanted s would be given as follow it's just Dirac spinners well as well let me introduce terminology this space s that's what feels is called Dirac spinners and uh it's chiral sub spaces in the even dimension are called are called positive or negative chirality well spinners okay so uh the spinners of that minimal vector multiplet in the dimension three is just Dirac spinners in the dimension four they are again Dirac spinners but they they split yes is s plus plus s minus because four is even in the dimension six uh one takes s plus times uh one takes the two copies of s plus so it's s plus tensor c two or doubles the number of uh while spinners so the two copies of while spinners of of the same chirality yes and then dimension 10 uh one takes s plus so they're just while spinners okay we are working again as the rest of our complex numbers if uh want to work over our real numbers then there are certain Majorana conditions that have to be imposed on on these spaces but to to understand the lecture you don't you don't need to uh stick one uh so let me write the dimensions dimension of s okay in the dimension three this is just we can just write the test as complex spaces so in dimension three we have just two component Dirac spinners in dimension four we have four component Dirac spinners in dimension six uh we have a four component while spinners but we tensor that with c two so that's c to eight and in dimension ten we have uh well two two to five that's uh 32 and s plus takes half of that so the 16 components is tensor or Dirac sun while spinners in six tensor tensor two copies so the so the the result is of dimension eight okay uh okay very well now okay now let me let me give it as an exercise in a cliffholt algebra I'll verify that the following statement is true there exists uh by linear form and let me code of type one type two in even dimension and uh in my notation so it's uh type two only in odd dimension the bollinear form on s so that's from s times s to c and let let me call that bollinear form like that uh which has sorry sorry invariant of course invariant bollinear form invariant with respect to the spin reaction uh which has the following symmetry properties in symmetry type as follows yeah there are one two cases in even dimension and yeah but it is relevant to these four cases special dimensions oh no no uh that that relates to uh to not those special four cases this relates to arbitrary dimension yes this is this is just Dirac spinners yes yes uh on Dirac spinners okay when Dirac spinners says so uh with the following symmetry type well actually let me just a bit extend that the definition of the bollinear form namely if we have the such bollinear form uh from c times uh sorry from s times s to c then we naturally can construct a bollinear form bollinear form which uh takes two spinners and maps that to external powers of vector space v just by inserting a cleft algebra action of the vectors from v on spinners so in components you take you know you take two spinners psi chi and map that to the bollinear psi gamma one dot gamma i one dot dot gamma i chi with chi where here we take anti-symmetric combination of the generators of cleft algebra uh so the the the exercise to verify is that such bollinear form of type one type two in even dimension type two in odd dimension exist and it has the following symmetry type namely it's symmetric type one is symmetric for r equals zero one plus d minus two over two mod four and anti-symmetric otherwise so that's two three plus d minus two over two mod oh yes thank you thank you yes k k k mm-hmm uh yes and type two is symmetric for k equals one two plus the lower integer part of d minus two over two mod four and anti-symmetric otherwise so was it a three plus the same yes okay so the the the bollinear of type two it exists in even at odd dimension so here i have written it just in terms of the integer part of d minus two over two which might be integer or half integer but you take the the maximal integer below it so now okay very well so you take uh you you you so to find uh those spinner spaces of the minimal vector multiplet the this the straight s we need to find the the symmetric b linear form on s times s to the vector spaces and uh that that's the case of k equals one yes it's zero or one it's zero or one plus d minus two one oh sorry maybe i put brackets here yes okay but by that brackets i meant that i want to take mod four of that whole thing okay okay so let's take k equals one yes let's take k equals one and let's see why did we need to tensor in the dimension six s s by c two well in if you take k equals one and equal six then you find that both for type one and type two form and type two two billionaires on Dirac spinners s are anti symmetric anti symmetric at well i mean at k equals one when you map that to vectors so that's uh no good to consider that's not that's not suitable for the Poincare c-pro algebra and therefore we we can fix that and take s times c two and just two copies of s and extend the b linear form by taking what we had on s and just anti symmetric pairing consecutive so if you if you if you like maybe actually if you like let's tensor it by c plus its natural dual space c c check and here we have sp one symplectic pairing between c and c check and for minimal one okay for minimal vector it turns out that it's efficient to take the positive corollity spinners and apply this construction and that's what you recover as a super Poincare algebra for six dimensional theory okay let me give a couple of remarks about those lists of dimensions which maybe mathematicians could appreciate the the the the the the exceptions of those list of numbers as follows are there questions so far about the exercise between your form super Poincare algebra it's module of the spin of the spin but but but but as we define super Poincare algebra we just needed to have a module of the genot part and yeah yes this is but for for for the construction of the super Poincare algebra we just have to have it as a module of a spin exactly exactly gamma maintenance s plus and s minus yes yes uh so gamma maintenance s plus and s minus uh maybe yes yes yes yes yes yes they are reducible representations of the spin uh so uh okay uh so let's let's let's say a couple of remarks about about those special cases when the three spinner spinner rule is holds did erase it already but you remember the cyclic permutation of our three spinners bracket of two spinners and then cliffholt algebra on the source spinner vanishes let me write it again it's important maybe you recognize something what you're familiar with so you take and take bracket and that's super algebra yes i want say two uh didn't i denoted by angle parenthesis and then you act by the resultant vector using cliffholt algebra action on psi three and we want the cyclic sum to vanish so so what is it about well the reason observation which goes back to square square an old one i think it's kuga kuga towncent of 83 of 83 that uh those four special cases and the corresponding representation of the minimal vector multiplet it relates to the list of the normed division algebras so the minimal super vector representation or minimal super point career relates to normed division algebras it's just real numbers complex numbers quaternions and actinians and uh as well a few if you if you like another connection the typo typological one that's also the list of uh it's naturally the morphic to the list of our paralyzable spheres so that's s not oops s1 s3 and s7 so the spheres are just the spheres of the imaginary units of those division algebras of norm one yeah so so the these guys is they have dimensions one two four eight and uh these are the sphere and you have to add two to the dimensions to get that list of numbers so absolutely so so the dimensions d of uh we called physics n equals one supersymmetry yes and the dimensions d is you you add two you get three four six ten okay and uh and if you if you forget about spinners and just try to work in terms of division algebras then that's three three spinor rule translate to the statement that well the de cyclic uh combination of their uh associator in these algebras vanishes now the spinners the slanted spinner space itself can be identified as follows so let's for bold k uh denote one of those division algebras so it's one of r c quaternions or actinians yes and for the spinner space as slanted we take k plus k so two copies of k and as you see the list of dimensions fortunately i deleted it yes but uh yes i deleted it but uh you remember it it went like no no no the list of dimensions of s yes let let me let me actually let me put over there so s s was again c to 2 c to 4 c to 8 and c to 16 and in terms of Dirac spinners it was s s plus times c2 and s plus okay and uh so that's for r c quaternions and actinians and uh let me just not touch this blackboard to keep to keep it at mind okay so you can think about that uh spinner space of the minimal supersymmetry as two copies of the ground division algebras k plus k okay and uh then the clefort algebra or actually here i didn't specify okay now we are working over the ground uh field the real numbers and for this statement to be precisely true we uh want to consider the vector space v with minkowski's signature so it's r to d minus one comma one okay so namely for this signature there could be imposed the real structure on uh those Dirac spinners on those lists that you that you have over there such that um okay such that such that the the following is a morphism holds so s as a real vector space is a real vector space of dimension uh corresponding to four eight or sixteen for these cases okay so our clefort algebra action of of r d minus one comma one on k plus k so we should write it as just s two by two matrices with a interest in k and we'll write it as follows so let's decompose r to d minus one comma one as follows let's take r to d plus r oh i'm sorry r to d minus two yes thank you r to d minus two plus r to one plus r to one zero plus r zero one so these guys are space like and this one is time like and let me put coordinates here is y x and t and then identify this r to d minus two with our division algebra okay and then the action the the the two by two matrices that act on the space are simply the matrices of determinant are simply anti-hermitian matrices of determinant one namely you write these matrices as as x plus t x minus t y and y conjugated so this is times identity matrix identity matrix and this is the dimension of of k by k well so this this is the element of the division algebra and this is it's it's conjugated so so everything so everything together it can act on on the spinors k times k so this is the clefort algebra action of element v from the space r to d minus one comma one that we have split it v as y x t that's from r to d minus two that's from r one that's from r one and r to d minus two we have identified with the the division algebra yes yes yes yes yes yes by the way the determinant of this matrix the determinant of this matrix is just the norm squared of this vector v so if you compute the determinant it would be yes x squared oh maybe maybe i want to delay t x yes yes yes so it's squared x squared minus some somewhere signs are wrong so minus plus no i want to have it yes i i i i i i i think let let me swap t and x and then it will work yes it's just it would be like that t plus x t minus x then it will work so it would be t squared then it would be a t squared minus x squared minus y squared, okay, if you take minus of that, yes, so it would be the the v squared in r to d minus 1 comma 1 v is the signature plus plus plus minus. So v squared is minus determinant of that matrix. Yes, yes, yes, yes, just take times that minus this time that and okay. Yes, it came from this condition, on this one. Which is some algebraic calculation? Which is algebraic calculation? Yes, yes, I didn't show you, you know. Can I talk to the physical meaning of that? Yes, so the physical meaning of that, that, we are coming to that, yes, I'll tell you in a second. So the physical meaning of that is that there exists what we called minimal massless vector multiplet. So there exists n equals 1, Young-Mills theory. Okay, so some people who are interested in high structures are in the audience. Let me tell you another remark how that relates to to high structures in geometry. I mean it says some of infinity algebras and that stuff. It's an observation by Bayes Huerta of 2010, not very recent. The remark by Bayes and Huerta of 2010 is the following. Okay, so what we have? Well, to consider that the super space V plus is slanted in those three special cases, V plus S plus straight S. And then this is symmetric bracket, yes, which takes S times S, takes two spinners and gets a vector from them. And now let me consider the following map. We call it alpha from S, from S times S, times V to C, which takes two spinners, psi, chi and vector V, and sends that to the bracket of psi and chi, computed by this operation. Yes, and then contraction using the metric with the vector V. Okay, now you can think about such map alpha as a three-core chain in the Chevrolet-Ellenberg complex for the caramology of the algebra g. This one, the super-algebra g with coefficients in the trivial representation C. And the condition that this three-cyclic condition, so the statement is that this every condition is equivalent, that's just an almost optology, it's an obvious that alpha is closed, alpha is three-core chain, well, then another thing that Bayes and Huerta observed is that alpha is non-trivial, so three-core cycle, and alpha is non-trivial, alpha is not a J of something, so which means that alpha actually represents non-trivial element of H3 of just C, and given a non-trivial element of H3, one can use it to extend this super-parametric algebra in a sense of L infinity algebras by the term in the degree 2. So with that extension, it means that for that list, d equals 3, 4, 6, 10, and the super-parametric algebra v plus s slanted can be extended to the super-algebra in a sense of L infinity algebras is defined by Schlesinger's stash of 35, but some people are very well familiar with it. There is three brackets. Well, the differential is from that Chevalier algebra complex, and that to the algebra, which relates to what piece is, by one dimensional space. Well, it takes two spinors and one vector, and it takes two odd elements and one even element. In physical picture, it relates to what is called B-field, which is connection, which is like connection on your own jorb in super-strength theory. I'm not going to expand on that. It was a side remark maybe to get you more interested in that specialist of dimensions and specialist of Clifford algebra representations. That's how their importance show ups in physics. Okay, this time it's 336. Maybe we can do a short break with two parts of the lecture. Okay, so let's continue our journey on the list of 3, 4, 6, 10 dimensional spaces. So let's actually write down the n equals 1 super-news that I promised to you exist in those dimensions. Okay, for start let's consider just a flat Minkowski space to D minus 1 comma D comma 1, and that's super-porn curve V-algebra. Summary 3, so for V plus slanted S, where S is related to the normed division algebras by taking two copies of that. So what is n equals 1 super-news is the following theory. So it's the gauge theory. So the gauge theory means that you pick a gauge group, you pick G to be compact in the group. Slightly later I will consider theories on modular manifolds than just R to D. So let me write a modular definition. So you pick a compactly group G and you consider a principal G bundle on your space time. Generally by X sub D I will denote the space time of the theory. Principal G bundle on X sub D. So then you consider connections, the space of connections, let me put it as clearly A with the connections on this G bundle. So connections can be represented by connection one-forms. So let A be connection one-form. After you pick a trivialization of the bundle, which means that you have covering derivative operator D plus A. Yes, and A is the one-form valued in the algebra G of the compact V group, not a super-group. No, no, no. No, I didn't have special notation for their Poincare supersymmetry group. I just denoted it as an isometry of vector space V plus S. Now I picked a compact V group, not a super-group. The space time X D is supposed to be equipped with a Riemannian metric G. If you are doing the theory in Euclidean space, I haven't fixed signatures more generally. There is a metric on the space X D and then we have the Young Mills functional. Yes, sometimes I'm talking about V, but yes. Yes, we can do the theory in different signatures and we can do it on different manifolds. I just, because supersymmetry later I will specialize for V, but now let me just give more general definition. So let's X D be a manifold with a metric around it, not degenerate metric. Let's put it non-degenerate metric. Okay, so Young Mills functional as you'll know is the following. It's a map from the space of connections to real numbers. This is called direction. Yes, and that's the integral over the manifold X D of the volume form on it, which is constructed by the metric G, times the curvature of the connection A, the norm of the curvature of the connection A evaluated by the metric. Using the Hodge star, I can write it as F A wedge star F A. Well, then I don't need the volume form, but okay, then let me finish like that. Just norm of the connection F A squared, the same as F A wedge star of F A. And we need trace. Yes, so we need to pick also. We need to pick non-degenerate. Let me actually say positive. Positive in variant bilinear form on the algebra G. So F is two form valued in G and we take that with a bilinear form. It's understood here that the norm squared is taken with respect to that. Okay, that's Young Mills functional. Now, super Young Mills functional. So for super Young Mills, now I will specialize to X D is R to D minus one comma one. And I consider the super Panker algebra iso of E plus slanted S. And we want to consider a functional, which is invariant under that super Panker algebra. So for that, we need the space of fields to be extended by the fermionic fields. And now the fields in the Young Mills theory. And so this is this is the space of fields. It's that same dimensional space. Now, the space of fields for super Young Mills theory is the following fields are, well, that's connection, connections A, the same that we had before, connection one form A. And for the fermions, we'll put a section upside of the bundle of the spin bundle in the first presentation, straight S, the spin bundle S over X valued in the algebra G. It's a joint valued spinor. Depending on the dimension, it could be Dirac spinor, no, in four dimensions or Dirac spinor in three dimensions. In six dimensions, two while spinors and ten dimensions is just one while spinner. Okay, so these are the fields. And the action functional is extended by the Dirac action. So, actually, let me put also a coupling constant over there, one over G squared, G squared Young Mills, one over G squared Young Mills. So the action functional S of super Young Mills is one over G squared. This number is called Young Mills coupling constant, the integral of Xd, which star of A plus the Dirac kinetic term for the fermions. And we write it as follows. Thanks to the fact that we have a symmetric pairing, remember that from S to S, we had a bracket, which send us to V. It means that if you take two spinors, well, it's symmetric. So if you take one spinner psi, psi, it's something on zero. So here we consider psi with that bracket dA. Okay, that makes sense. Well, since we have a metric, it doesn't matter whether we send it to vector. So one form is symmetric, you can identify that. So dA of chi is one form valued spinner, and then with the bracket we can construct, we can construct it. So usually, in usual thesis notations is denoted by the slash of the Dirac operator. Okay, notice that these definitions, they make sense on arbitrary smooth manifold with a metric, which, well, is a spin structure. Maybe I should be more careful and just say that we want the manifold. It's sufficient to have the manifold equipped with a spin C structure, not necessarily a spin structure. And that means that there is an extra, you want to actually the fermions, but then the equations could be, I mean, the young mill's functional could be written down on arbitrary manifold with a spin C structure. So far, we've just written the functional, but we haven't said what's special about the function. Well, the special thing about the functional is that, at least in the space time, if the flat, if the flat I find space, we can make that functional invariant under super angle mills on r, d minus 1, 1, is invariant under super point carrier, the algebra, isometry of v plus straight s. The super park rail algebra, it takes some fields. So the translation generators and rotation generators act in a very familiar way. I don't have to write the action of those generators. The odd generators, which are usually called q's in the physical literature, so let q be generators here. And let's say that an element, I will write an element of s as epsilon on some basis in epsilon alpha q alpha. So let's say that q alpha would be bases in s. So q alphas are odd in my conventions, and epsilon alpha is even. So epsilon alpha is even element of the even space of spinors s. Is this called general ideal or only fp-doubts version case? No, this is in a special case. The n equals 1 young nose functional that I'm going to write down is for the special case. So that's for d, 3, 4, 6, 10. In general, I'm trying to stick with the notations that straight s denotes that special case. Okay, now the transformations. So here are the transformations that the action by q equals q alpha epsilon alpha, where epsilon alpha is a spinner in a straight s, on the field it goes as follows. Delta a is one half lambda gamma mu epsilon. So let me write it actually in a clear for the notations, but I think you can translate it to the notations of the odd bracket that we had before. Well, what stands here? Here you take two, I'm sorry, how did I call it psi? I'm sorry, I did call it yes. So psi is the field that you have in this theory, the fermionic field, the super partner of the bosonic field of the connection one form a. And epsilon is a parameter which generates a supersymmetry. Okay, so you take these two guides and since you have a bracket from the two spinor spaces to the vectors, you can dualize it to get a one form, then you have a map from psi epsilon to one form, so that's the deformation of the connection one form a. Components, write it like that. Then the variation of the fermions is the following minus one four. Here is you take the clefert action of the connection one form f on the spinor epsilon and that's it. The clefert action in the components, excuse me, that f a is a two form, yes. So you can act by clefert on the representation of clefert algebra, so it's gamma mu on epsilon, yes. Sometimes people use slash notation for the denote detection of the clefert algebra. Okay, that's it. So that's it so far. It's called on-shell representation of n equals one vector multiplied and the words on-shell means the following. It means that the Young-Mills functional as of super Young-Mills is invariant under the supersymmetry action up to equations of motion and equations of motion can put it as ds. If you think about s as a function on the space of fields, then the variation of the super Young-Mills under the supersymmetry action is 0 up to exact, sorry, is 0 up to ds terms. So which vanish in the critical points, the critical points for physicists, the equations, the dynamics of the classical theory and this slang on-shell means that for free particles they satisfy equation p squared equals m squared that's called mass-shell. It's a bit outdated terminology, but we still are using it. Okay, now let's consider the following situation. Let me take a spinor epsilon and consider the supercharge q epsilon squared. Okay, so by a superpoint carrier algebra it generates a translation in the direction of the vector field v which is bracket epsilon epsilon. Isn't that that bracket in terms of gamma matrices? It's epsilon gamma mu epsilon. So you call the vector value delinear, if you wish, delinear in epsilon is a translation along the vector field v. Yes, yes, yes, derivative along the vector v. So roughly speaking, let's see how it works. If you take the field a mu, yes, you act by q epsilon, you get that. Now if you want to compute the square of the q epsilon, then you need to take the variation of psi and when you take the variation of psi, you see that you get a curvature here times, well, two gamma matrices times the third gamma matrix times epsilon and then after a little manipulation with the cleft-algebra, so using that axon of the superpoint carrier algebra, namely that three-spinner rule, you will find that the delta squared variation of a mu is the lead derivative of a mu and that's IV. Let me write without indices. So that's a contraction of the connection, sorry, of the curvature two form is a vector field v that gets one form. This is what variation of the connection one form is and this is gauging variant lead derivative along the vector field. Okay, similarly for spinors. Now let's discuss the construction of dimensional reduction. So dimensional reduction means the following. You start with the theory on the manifold xd and consider and take xd in the form m over d prime times v of d minus d prime, so where d prime is less than d and you compliment that manifold of lesser dimension by just a flat space, so that the total is of dimension d. More generally you can consider a vector bundle over m d prime of dimension d minus d prime to construct dimension or reduce theories. So this dimensional reduction means that you consider that you write down your fields and action on the manifold xd and then take all fields to be invariant along the translations in the flat vector space of the fibers of the vector bundle and so you get the theory which depends only on the fields defined on the manifold m d prime. We will call a dimensional reduction from xd to m d prime for such a product or more generally for twist a product. Q sub epsilon. Yes, yes, yes. That's Q sub epsilon. Okay, so now let me consider the following station. Let's consider dimensionally reduced theory and let's pick Q sub epsilon for that dimensional reduction. Dimensional reduction when you take xd as m d prime is vector space of d minus d prime and I want Q epsilon to square to the lead derivative by the vector field v which has only components in the vertical direction. Let me go for the pictures. Think about m as horizontal space and vd minus d prime as vertical space. The fibers of this vector bundle and I want the vector field v to have components only along the vertical direction. Okay, so in the dimension or reduced theory it means that dimensionally reduced theory on md on md prime Q actually squares to zero. Since it squares to zero one can consider co-homological theory with respect to this odd differential and this is what the station when Q epsilon squares to zero is called, I think it goes back to Wheaton in 80s co-homological field theories. Well actually in a sense it was earlier that the earlier constructions which really rely not on supersymmetry but on what's called QBRST operators for the gauge invariance functionals. You can also be realized in a form in a co-homological form namely there are in the QBRST the gauge fixing construction by what's called BST complex Q squares to gauge transformation and then if it is extended properly by auxiliary fields it squares to zero and the co-homology of the operator Q are a gauge invariant observables in the theories with gauge symmetries. So this construction doesn't necessarily deal with gauge symmetry you see it deals with the superpunk or the algebra is dimensional reduction and so it applies not only to gauge theories but to other stuff. So let's now take that list of examples 3, 4, 6, 10 and see what we can do using this construction. So you remember the decomposition of D to two-dimensional space plus the space of division algebras and that's what we will combine with this construction of homological field series to get some nice homological field series as follows. So XD would be the space in the form MD-2 times C or R2 so that's two-dimensional space. Now I'm talking about real smooth manifolds but I like to identify this two-dimensional vector bundle over MD-2 with a complex with a one-dimensional complex line bundle by the following way. So we know that Q epsilon squares to the translation I mean no we take we pick a suitable spinner epsilon such that it squares to the translation only in vertical directions is a long fibres long fibres R2 and another important point is that if you are that V the norm of the vector V is 0 that's always true I mean the general statement about n equals 1 supersymmetry transformation and it follows again from the three cyclic spinor rule so V is epsilon gamma mu epsilon and if you can compute V squared it would be epsilon gamma mu epsilon with epsilon gamma mu epsilon and since you can cyclically permute that and it's still the same expression the sum of that vanishes so it has to be 0 by 3 spinor okay so so we always have the lead derivative along the null vector field and in Euclidean signature it has to be a complex yes and so if so then then an insutable metric on this R2 here mission metric on that line complex line fiber is just D over Z bar where Z is a coordinate is vertical coordinate so now I'm taking the Euclidean signature Euclidean signature so it's Euclidean signature here and there as well that's why you have to introduce a complex column so V is a D over Z bar and since metric has only components between Z and Z bar yes the the normal this vector field is a is 0 so V squares to 0 okay okay very well now recall that we have a presentation of the superponkary leal algebra using the decomposition of D dimensional space into two-dimensional subspace and D minus two dimensional so that would be D minus two-dimensional that would be two-dimensional and we have identification of the tangent bundle you can identify a tangent bundle of the two D minus dimensional space VZ is one of the real with one of the normed division algebras so which has dimension 1 2 4 8 okay and constantly the supersymmetry just formations and then analyzed with respect to this structure and it gives the following the questions let's continue okay so under the dimensional reduction to the D minus two-dimensional space we still have the same number of components of the spinors of this of the spinors that I called psi which are sections of the spinner bundle straight S however from the components of the gauge connection on the space X D we get the gauge connection on the space M D minus two and plus two skillers which correspond to the components of the gauge connection along the two vertical direction so after such reduction a on X D gives the connection a on M D minus two and two scourer scourer fuels usually called fine fiber which are the vertical components which are exactly from vertical components from a along the vertical components called the V dim D minus D prime and that's what we've denoted over there just as a C or as a okay now now one one one one can analyze the so such dimension to reduce and cause one young mills and on the space on the space of these fields with the fermions and deduce that the integral that the path integral or the space of connections is in the space of fermions in the superannual section it localizes localizes in a sense of I t a boat but applied to infinite dimensions so the infinite dimensional station was here is discussed by Witten and then in the math literature it's I think by paper by I t I t a Jeffrey so it's not well it's the statement that semi-classical limit is exact yes it's the that that that holds in that situation of the supersymmetry which can be interpreted at a certain homological computation namely since namely we think about the differential Q which squares to zero as a as a differential in certain complex of fields and we think about the volume form which we integrate in the path integral as a SD sorry the Q closed form and then one can localize it to well one one one can localize it to what this is called the BPS configurations mathematically there are sections of a certain vector bundle over this piece over the space of fields sorry zeros of section of certain vector bundle over the space of fields and those sections of vector bundle over the space of fields what's this is called BPS equations so so so the finite dimensional cartoon of this localization procedure is a point career hope for someone else's theorem that the integral of earlier characteristic the the other characteristic of a manifold which is the integral of the early class of tension bundle can be computed by picking just a vector field which is a section of that vector bundle and counting the zeros of that vector field is science so that's what happens here so it localizes to the integral when you write it is a D dimensional version of the D minus mm-hmm yes yes yes so when I write here a is the connections and those yes yes yes yes so this is K value 10 this is just colors yes yes yes yes so okay so it localizes on what we call BPS configurations namely it's a certain modular space it's the modular space which let me denote as M the modular space that the data of that modular space is the space time manifold the reduced one the chase of supersymmetry transformation to construct the differential Q the chase of the gauge group G okay and and and here you would have well if you have some observable here let me do it with an observable the restriction of that observable to the space of such BPS configurations and that observable here it satisfies that Q epsilon the observable is zero here it satisfies just the closeness condition that D of or is zero so you integrate just a co-comological class in the DRAM co-homology on the space of BPS configurations now concretely DRAM differential so all is a differential is differential form on this space just in the DRAM complex on M and D is DRAM differential okay so the BPS configurations they come from this analysis so remember that you had the equations that where the variation of the fermions is f which x by Dirac which x by Clifford algebra on epsilon and now let's consider this equations this is the BPS equations I mean the the variation of fermions is zero is what we call BPS equations so delta is not zero no delta of a delta of a is not zero so remember that delta of a is the gauge transformation along the vector field V so here it will be just the contraction of that vertical vector field with the f which is obtained by the reduction of the high-dimensional theory so for for delta a it will just give us the variation of phi if I think okay if if I if I've taken the convention the delta squares to the lead derivative along the vector phi so that's gauge transformation by phi so F menu gamma menu epsilon 0 okay now this analysis is in D minus two-dimensional space and you think about epsilon just an element of K so the you recall that S is KK and pick for epsilon something from here I'll take epsilon to be for example 1 0 component then Q epsilon squares that you can verify only to the directions along the along those matrices t minus x t plus x and here would be 0 so it squares so it's vertical however okay now using the the Clifford algebra units of the division algebra and so this equations tell us the following that the spinner epsilon is invariant spin tangent to the manifold m with D minus two minus two action is generator F menu yes so F menu is an element of the external of the external square of the cotangent bundle of MD minus two and so yes and if we're forgetting about the le algebra yes so so epsilon is not the le algebra extremely on epsilon I mean it's it's a long it's along all components of the le algebra the equation holds and and that it uses to the following list of equations well so let's now write down our favorite list so the here I will put D of the XD 346 10 here we put MD minus 2 it will be one-dimensional theory two dimensional theory four-dimensional theory and eight-dimensional theory and here I would put equation f slash epsilon is zero well in the one-dimensional situation you have just so to action and so to x and epsilon with a single generator F12 and oh I'm sorry I'm sorry I'm confused in one-dimensional station there is just no two forms the space of two forms is empty there is nothing to rotate so in one-dimensional station there is no curvature and the space of equations is just empty then in two-dimensional station there is a one generator of the external square of the cotangent space and that generator is SO2 generator and it acts on epsilon so the the equations are just that F is zero so here I'm putting F along of course this is the curvature along MD minus 2 yes the curvature in that really reduced theory so F is zero okay in the four-dimensional station okay in the four-dimensional station so the space of two forms let's say of R4 it decomposes as the space of a self-dual two forms plus the space of anti-self-dual two forms and epsilon is acted by the SU2 you have corresponding decomposition of the spin for as SU2 times SU2 so epsilon is acted only by one copy of that SU2 well depending on our conventions where we've put epsilon here as a corralty and therefore that one corralty that is one part of duality of the curvature tensor has to vanish so the equations here are F plus zero where the plus is defined with respect to epsilon okay and finally for for the actinionic situation well here we can write equations in terms of epsilon I like more a different more differential geometric presentation namely this equations can be it is to compute that the consensus of this equation is the following define the the four form omega by the bilinear by the four form bilinear so it's gamma mu 1 and me to gamma 3 and me for epsilon yes using the cliff or algebra action and the pairing between the epsilon spinors so this is four form is automatically self-dual for form and the equations can be written as F wage omega star is equal to F and this equations are called the spin seven instantons these equations are called just instantons so it's been seven instantons and a dimensional manifolds instantons and four dimensional manifolds is called just flat curvature and here we haven't you have empty equations so this is the list of the nice the homological field theories that you can construct by a dimensional reduction of n equals one three prime means by two dimensions we had like a ten minutes break so maybe I'm finishing in four minutes okay so let me let me just write it again definition so let m d minus two be smooth Riemannian manifold and let me define the modular space m of m epsilon g as follows is the space of connections a on principle g bundle on m d minus two so g would be compactly group yes which satisfy the equation p epsilon of f is zero and p sub epsilon of f just denotes those equations yes just just these equations so in all those cases they are given as force and epsilon is fixed yes and epsilon fixed it means that explicitly it means that in the four-dimensional in two-dimensional case you have to fix nothing four dimensional case you fix what you call self-dual forms that you call anti-self-dual close you fix orientation in eight-dimensional case you fix the self-dual form not degenerate self-dual form which rise to actonians by Kylie yes so p epsilon of f is zero and mod of the morphism that's gauge transformation of the of the g bundle so if you count the components the number of equations here well it's naturally from here they just identified with the imaginary units in this normed division algebra so there is a one equation here three equation here seven equations here and the number of components of the connection is has one more component but there is a gauge transformation so overall if you do infinitesimal analysis of this modular space you find that the equations which define m are elliptic equations so this modular space is defined by nice elliptic PD and so for that and so it's finite-dimensional and there and therefore it's fine dimensional yes and therefore it's fine dimensional so now we are coming to the modular spaces very that I very liked by mathematicians and geometers for example take a two-dimensional manifold to be a human surface and here we are talking about the modular space of lead connections on the reman surface in four dimensions well if you if you take four-dimension manifold to be complex manifold then instantons can be rewritten as a holomorphic g bundle so there will be the modular space of homomorphic g bundles on complex two dimensional surfaces well in that dimensional it's something more exotic all people are working it but I'm not sure if there are some well there are examples but there is no more or less general theory for spin seven instantons but well they exist so this is the that finite-dimensional that nice finite-dimensional modular space well in the generic point it's a smooth it's a smooth manifold but because of because sometimes that this action is not free there could be singularities of this manifold so if you you you you you're supposed to understand it in a sticky sense in geometrical in geometrical approach ok and now the the statement about which I'm finishing is the following so let me just write that write three lines here and finish so the three lines are the following you consider if you consider n equals 1 super young mills on the manifold of the form m d minus 2 reduced so ok reduced reduced n equals 1 super among meals to m d minus 2 m d minus 2 times this one or m d minus 2 times elliptic curve so here I mean the complete reduction in a sense I've defined before here I mean taking the manifold to be d minus one dimensional and reducing only one direction and here I'm taking the manifold to be just a product of elliptic curve for Riemann surface of genus one or finite size and not actually reducing but just considering the theory on such product now here q squares to zero here q squares to the translation along this circle s1 and here q squares to the operator or an anti-halomorphic vector field on elliptic curve is uptown and correspondingly the space of observables so observables in those three cases they correspond to the following observables are in terms of the final dimensional geometry of the module spaces so in the first case they are just diram caramology of the modular space m jm epsilon yes in this situation we when we have added one circle we need to consider the fields which are loop valued and that means that you take the loop space of m and you consider the equivalent caramologists with respect to this action so it would be diram caramologists equivalent with respect to the circle action on the loop space that's infinite dimensional space so let me put it as m to s1 and that's the loop space and that's the same as case theory that the same as case theory of that modular space and finally don't have space here so let me continue to here in the elliptic case we would have the caramologists equivalent with the dz bar operator on the elliptic curve on the space of maps from the elliptic curve to the modular space m so let me put it like that so physicists call it chiral chiral model or zero to model or written elliptic genus elliptic genus there are many names but the the convenient name for that is elliptic caramology with respect to the elliptic curve is parameter tau of the modular space and gm epsilon so that's what the n equals one young meals on those manifolds give access to it gives access to the study of this geomological theories that there there is other names are rational trigonometric and elliptic as we will see in the connection to the integrable systems so let me finish here for the first part