1/3 Supersymmetric Vacua and Integrability

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1/3 Supersymmetric Vacua and Integrability
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"I review the relationship between supersymmetric gauge theories and quantum integrable systems. From the quantum integrability side this relation includes various spin chains, as well as many well-known quantum many body systems like elliptic Calogero-Moser system and generalisations. From the gauge theory side one has supersymmetric gauge theories with four (and eight) supercharges in various space-time dimensions (compactified to two-dimensions, or in Omega-background). Gauge theory perspective provides the exact energy spectrum of corresponding quantum integrable system. Key elements, usually appearing in the topic of quantum integrability, such as Baxter equation, Yang-Yang function, Bethe equation, spectral curve, Yangian, quantum affine algebra, quantum elliptic algebra - all acquire meaning in the supersymmetric gauge theory."
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1st lecture and during 1st lecture I wanted to focus on supersymmetry since my electors will be vindicated relations between supersymmetry and interoperability I have to review both parts of 2 days the mostly doing supersymmetry and next
lecture I believe problem was that going into the ability but there will be some to also and selection some supersymmetry so I want to start with supersymmetric so there is no reason for the charge and I want to think about the salaries and Clinton dimensions In 1 must wonder mentioned are called and you will too I want to thank for for attacking I want to think about two-dimensional space to be sitting so this is a circuit and this will be tightened their action but later later the level of discussion when this is arbitrary arbitrary mention surface with generation and punches and later in the later there will be some dimensions for international flights and maybe even when 6 dimensional salaries but which I will think as two-dimensional salaries with infinitely many feel that may already here say the words about 2 what I mean by this and for what most of you know I actually designed this lectures to reverse the mentor but I think that the sinks that I will be discussing the planet expert myself but they still tried to make it to man so when you think about this look like if you are in the same dimensions you can't think about three-dimensional space to be but I don't hear consists of some produced and that's two-dimensional salary viewed in terms of the celery on on this 1 will have infinitely many fields which are usually called culture-clash mall and they will be masses and masses of proportions to close climb on the way if you are 4 dimensions you can begin to think about 4 international salaries as stood horrors and gain come to also will give the infinitely many fields In this two-dimensional salary on cylinder but there is an alternative when thinking about the higher-dimensional salaries to be lower dimensional and this is achieved by Omega background the only thing will need from here since I assumed that the where familiar with cold filariasis is that of the of Coast clients over the circle there is a limited to the fact that we consider the identification of the space coordinates along with thinks are 1 which will make us so we identified coordinator here that's supposed to be I wouldn't if I can exploit and this is the shift so we're using the transformation from translations and that leads us to go to company occasionally hysterical and only about the only thing that actually we have to to think about this and instead of such identification with are let's say it is hard to mention that instead of Tito would take part when you rotation what which recalled and so this kind of think I will call instead of colors going to excellent and it's actually a simple thing to see that the from the point of two-dimensional salary when replaced futile while by Omega background was that they think this will be a game distillery was infinitely many fields and masses will be related to the mall itself said expansion in spirit of fun so that's what I said here is that the West having now for superchargers dimensions but this story is not necessary unnoticed two-dimensional they can become a decline from the dimensions automation actually fight dimension or Omega backgrounds coming from 4 dimension this requires 2 space real dimension as many months so we can think about Omega backgrounds coming from 4 dimensions or 6 dimensions if we have to angles like Absolut so this is a set up and now I want to say a few words about the algebra also for superchargers so 1st admitted that space of vacuum I need to explain which means supersymmetric work In salaries ways for charges carries representation all of completed associative algebra which is called in using that old attack this algebra is called twist kind so this is something that were 1st interested then let me introduce object In order to introduce that we have to remember that forced the charges it means that we have let's look let's labels and Cuba last Q minus Q plus bar and Q minus bodices justification of 4 charges and we need all the nontrivial and accommodation commutation relations few plus-minus the antechamber character was Q plus-minus Bob 2 times have gone and plus-minus moment so this is about to rest this stuff is and if we want to write comment on any of us want to cancel his plus-minus translation we can introduce tool In the combination to independently in the combinations are the ones with the related to want to make with some coefficients that Keyway will be denoted by Q plus-plus Q minus by and QBE dispute plus plus Q mine so we this notation you can check the Hambletonian yes and QA square equal 0 and square equals 0 as a consequence of supersymmetric algebra and you can checks at 101 harmonic involves we can write here in part to the Cuban and I was just harmonic in "quotation mark comparing applause being beaten .period plastered on duty and these costs of their own soldiers just for a moment so this is Hambletonian and now the power of the supersymmetry was forced the charges is when all these things I'm talking about goes back to Sweden in later years and very well used and described by chocolaty and buff also in their nineties the region probably late eighties and surgical too often and so is a sad that statement that the
size of a questionnaire asking his excuse questions what is a cohomology all of so I will pick 1 of them conspicuous and we now just noted by Cuba the question is what our ecology classes all a of the matter I picked so first-aid mini-games supposed son such costs surgical which means that the size state in our quantum system which is having the symmetry the in history such campsite equals 0 a flight is not I had pleased and I get a function of time and money and with no 0 eigenvalues or you need a combination of souls the number of people who are on still control the plan is actually used to call home and when show it just right pairs you could argue plus a few dogged him or which is the when talking about the size of his kid like you know this is 0 so it's excuse of queued under because thereof and we got the statement that the we can write that as killed called 1 of here that which means that the Securities Act and its through commentaries on for size that spanned the no 1 0 I didn't group functions of monitoring and the use of 101 this is the single most which those only nontrivial cohomology classes part corresponding to rely on the so we are now discussing where just saying that the lock you which preserves supersymmetry is approximately backyard maybe at this hearing ,comma logic losses of Q so that's us the statement so we assume that was at a job site because so now let's multiplied left pipe and calculates that this is a place where I am using Norman In this discussion don't have to use and along with the approved the statement I have to use the Northside seems to think it would themselves a vacuum that uses color products so there is from the definition of age is nothing more without the to square of Cup's I'm model growth plus you've that your side of the square because age is look less kind of operators and this is equal to 0 from which I conclude that assuming that's the norm is the dinner at the Cup's I going 0 thank you that closed so we choose that stating that the plan which is what you are harmonic representatives Of course now there is 1 more think you have to do to prove that there is a one-to-one correspondence between supersymmetric Lockyer and cohomology is that if you U.S. you as the site so it's too far it do and the sigh I function of 101 and withdrawing and you have to did deal said subsides jail or it's actually 0 a simple calculation shows led a plan of that is equal to so it's a chance is not there you can't do it that I advise you something which implies that is the simple proof now if if you assume a child from 0 it implies that killed on Friday called on from which closed again so I could I thought I just think explained is that the there's a yearly one-to-one correspondence between Q :colon colleges in L'Aquila and if we want to count supersymmetric Walkure currently we can ask the question the described chronological assistive so the statement was independent of choices color product except to proving that its monarch representative so now and I think I want to go from states 2 the suppose all I see is you cohomology is an opera which means that security so I could 0 up to the issue of 204 last sometime that important thing and this is what I stated over there the space of what the presentation of commutative associative algebra is we've always and all times Folger and this index takes values somewhere and we will actually tidings where takes so I O J also compared with 2 tutorial calculations so we can write that the knowledge put acts here the they can be in any position when X and Y are recording here we forgo the sex and so on not to mix with the other thing with important 1 that's all I is a function of X
laughter is independent Of course molecule and this is consequent to the fact that the fight they can shave ex-wife that exceed the 5 moves point that this will be all viable Lescure something and the reason it's kill something is Bacall's shift operator which is please can you return as a accused something from this formal so things X plus there is the same here as if I would probably think is the same as all 5 plus time please taking 1 all I offer a few times all is it means that the all Texas independent of positions this takes its formal offer few commentators results and conclude that's all I had no idea drops independence .period time can we arranged in terms of the some of the which also has a properties it's incumbent was accused now these indices around in the space of all operators which competes with last year and the statement is the 1 that I wrote in that side of it's symmetric and I replacement White Jr and this is because we are in because we're in 2 dimensions or higher this will not be true in 1 dimension because on the green in 2 dimensions and higher makes no meaning since anywhere I put hear an 6 and what which I in the press I cannot make experts around anyway but I can't say what you order because dependence on position of these guide with its 1st or the 2nd but is is a dependence on eggs and this 1 I already proved that 2 Our to exact and things include that relations but the guy is semantics so we have no systematic metric CIG where indices Jr care in every single using belong to them labeled by his where is let me explain this last statement we can work something interesting the ball without such 1 operator there the or listen differently support you found 1 vacuum which is supersymmetry let's call it all but we have to discuss the uniqueness of this 1 states that week among the spotlight today advised to read the will of the pirated goods and waffle from 1992 suppose you find wants special inspections of what you have to find his own son and his wife 0 or equivalently Cup's quality it was my son I didn't was signed last year what defines such once and now we have the operator or why which commits with Q reform others which is all I can think of obviously this also be belongs to like him because if you will and who later the seems to want to commute result so now we found as many vacuum states as we have operator saw so there is a statement that there is an operator state correspondent which will be be there are some subtleties which might be related to the structure of this constant educate but when I will move to the political field service I will discuss subtleties but In a generic situations there is complete operators state correspondence and they had to prove history and now we asked to type of questions 1st question would be the it wouldn't have presentation of commodity Ringo went from West Inc which show that the operator of the state which is a 0 I guess defined now that the structure of the Vikings all this for this next the went 1st to think we needed what offered the correspondence and sometimes fails but where engineers now are discussing what kind of structures in space Hess for that quarter thinks 1st we know that it's representation of succumbed to to what kind of representation with were will be discussing entire enduring interest in this league but is aware that we give it a name if I did not give the is called twisted Kiraly so what we have here 2nd that we have this is the state of ocular described by the states and it's very simple to show that operators or let's call it all can not diagonal basis which means that all I is not a good idea director Of all just if it would make investors and just stuck in said that acting on it and it the problem you almost coming that's what I said in there the matter because that's the idea is mandatory you were here when there are going to it's important I what to remind that about more important than this and I would like to hear if the objects of accommodative arrangements so we have a big and we have this basis and as many commuting operators which is set of note classical version of this quantum-mechanical statement would be we have a face-to-face which is the replacement of the space of and we have passed on commuting operators and exactly as many as half of the dimension of a face so ,comma conversion of is called classical intolerable
system classical version of the space of some dimension to an and and commuting costs on coming but because of Dole's age I want to and so this would be a classical but where quantum worship and this is quantum considerable so statement that was the space of ocular supersymmetric gage the supersymmetric sewers into dimensions was forcibly carries representation of committed to assist at the algebra the same statement as and the operators follows giving us continent available system and this a generic captured the space of action with 1 2nd once so space of IQ will right last definition completely identified with a common knowledge of self-protection that that's what we'll be doing to my lecture that's lectures about OK so that was good news here and 1 not so good the good news is that the way made very general statement but if you study space of supersymmetric What You Are you might something about quantum interoperable system but what quantum into global system you have I say nothing about the with I have to identify what condiment audible system it actually the space of supersymmetric back you don't have to to be doesn't have to be funded dimension there are many simple examples you can give that suspects of suicide can be infinite dimensional but certified direction and that set starts what what do you mean by this but when it's finally dimensional Hilbert space there a very Woods said and you can ask the question of what you just ask so in all I've mentioned here people budget cuts in half what they were doing In those old days was a what's called a vacuum so let me give you some flavor with vacuum bundle laws and I want to formulate all the questions and then show why knew newer questions are interesting and new questions kind of started with the paper good morning than aggressive and I rode in the mid-Nineties a few years after that to quote the office the old days what they would consider it tank space of parameters of your um and equal to the "quotation mark quantum field theory it comes with some number of parameters let's call this basic parameters local according to and point here this is some point in the space of 2 varies space of right because they like you it is quantum field theory in 2 dimensions that can maneuver parameters the space can change so you register parameter and now think about this as a vector spaces and you have some part of transport you're moving along base so this entire total space they called the vacuum please comes with some connection let's call novel a local court answers since the denoted his likely to and what you got in Lahore showed there these connections if deformed by when you take a stake here you connect on that status they said but all I can create another about you so you can actually combined was arbitrary conditions In this connection that you hadn't there in the bundle plus some coefficient times the actions of the operator this could be a connection between them and they show that the flag so they would stand In this sense so many thinks about this flat connections in the space of Viking bottles for example they could have started some equations like seasonal totals the flatness conditions called start question because it's flat and there is a conjugate 1 and the commentator of this was going to get 0 and the conjugate 1 would do differentiation in accomplished parameters and so on and then there isn't our business of thousands of topological quantum physics salaries and do in my lectures will not be about this although if there is some interest I can connect my lectures to stuff especially in December there was paper by the skin chocolatey and a bonfire on Kyodo News care needs needs cage according where in the language of the vacuum bundle into 2 start question their interpreted sinks will be the main part of my lecture so in principle I might get back to this explaining how these guys relates to 1 of the OK now 1 important question that everybody who works in quantum interoperability asks so was this assumption that there is unique stating the place of from which was tough to construct the king's operators and we don't have this unique state we cannot have operator state correspondence with described for example you this is not 1 state but millions generous state and so on we're kind of interrupt identification but that they can put enough of you to arguments 1 argument was if the salary is super conformal in this area comes from so conform which means what these algebra but the usefulness of the bunker algebra isolated tracts of conformal to wrap that there is something called you 1 cup in any closer to a super conformal Algebra Games is part of my job but I need the argument many quilters super conformal under 1 kind and as such there is a spectral floor spectral fall is the relation between Nearview France and Ramon sector this over supersymmetric and there are others warrants for someone who did not that is the subject to the new slots in the mornings at Fermi sensors and firmly on the can ask questions what kind of boundary conditions you put I'm what kind of boundary condition you put on the family and it could put periodical Ontario so you would have to build a boundary conditions were talking about the fact that the interview with periodic boundary conditions you are talking about a month after the slack you I'm describing guiding them on these vector 0 is in government sector because I put the boundary conditions me I have hidden somewhere here but if you look on the government's so if you want the state to be and the furious super conformal set up then it's extremely simple argument what you do you
take him to sector and it goes was bumper-to-bumper condition and there is not much of a new start sector because of our belief that the unique clerk and then you want current which relates a staging area reflects tool then he stated on 1 sector by what's called spectral falls in order disguise that you have to modernize the 1 parent and construct the operates the operator and then spent operator allows you to connect state actually it's mostly coming to devastating floods sector is emotionally connected to this so if this is not fight sectors resisted flock here and it is common sectors then there is specter of from 1 to another and gives you state is very important it's control of the that all piece of his being operated the operator has unique all peace it was any this is important because you take the state Munich stadium units in the salaries last if it's not super conform so this was about separate components of conformal that is unique state which leads to this theory from 1 sector and this is a state which comes respectful fall from this meaning the actual guy has its own way to describe it that see if you have super conformal celery that have left wing sector in right moving sector and you have you charges which left we won and right moving so every state in a vacuum in 1 sector will be labeled by 2 numbers now mathematically This is a statement about that see you can really or in a complex set up your back described kind of in In the in the language of the complex geometry and then you have .period Q for it's not only that the details of former can count but you can call the homomorphic and home-office part of the degree of the and that's the region that they can so these guys and is the 1 which has as a difference equals so this is a minimal 1 now if stories massive there is no you want kind there is also a conformance with the lets feels that is identification between in monster and it goes like this yeah .period unsteady conditions it would abstain operator it'll make it so operator always exists can go from the periodic the boundary conditions and these guys identifies everything here we hear and interest it is actually true in an equal 1 is that growth the cannot find I explained in 1994 for an equal 1 salaries which are also related to the special German to mindful of their is no spectral floor but this is kind of the spectral jobs for every state in the months that 30 stayed in the reflects and way so we take this guy which is a minimal 1 and on government and there is a correspondent he said he was not on the of drawings that ongoing but Lagrangian and Hambletonian they said did you come from the beginning of the lecture I said that it will be extremely primitive which I will do everything we wanted up so I said that it will be extremely Elementary and only then alleged except quantum mechanics it's important fields or some kind of calculation the knowledge of calculations of some interesting so anyway so they I argued now quantum-mechanical languages and with using fermions was to come to condition that there is always unique status as a reason for supercharger and this unique state comes from the vacuum of shots so I'm using this to use the operator the correspondence now I'm in a business to ask the question why don't I take analogies all licensed operators which forms the commodity love run and find common of these other so I'm asking the following question 5 Coleman instead for all obviously this statement is currently being cohomology so the statement is me I functions of the old is commuting also good commuting operators have to introduce on that basis of seals and that means that basis of and I know how to introduce that I spend some time arguing that the policy labels read common states in the statement that I will find energy spectrum common eigenvalues and I states all of these all which is different than the topological quantum field theory stuff when it says take guys you have the flight connections diagonal CIG case because its symmetric and then calculate correlation functions of the operators and right generating function of the correlation functions of walls which is off the world correlation function like that and :colon so this was a question of topological quantum fields I'm asking different questions to find a common ideas and why am I asking this question because the statement I made that way have a continent is that this is a a natural question quantum integral to describe exactly they will find the nation as I said there the paper I wrote was an aggressive recently which led so if there are no other questions I want to go to the action functional space so fields but the idea of what the parameters to give some feeling with salaries more questions than answers the rise of the usual questions on the matter but if I find 1 said I can find another set of accusatory up so I have to take such a step and I will be taking in some clever OK so if you wish that I had here Porter things it's this to type of salaries 1 which comes from coal declined directions another which comes with this kind of I want to emphasize important difference between this and the salaries don't break Lawrence in light of the service of Lawrence environment and higher dimensions the salaries will be Lawrence warranty law dimension but the rotation was excellent will break Lawrence environs in higher dimension because if you take this please this piece of art to your salary is cylinder the Times is Ottawa's Epsilon and the absolute rotation broke Lorenson oriented the film so high emissions so rule-of-law insolent lowered emissions will be creating the salaries In them lower dimension and in this particular case somehow cleverly the knowledge of some distinction high-dimensional story will will tell me me how to fix basis in the law admissions but this question of what is the right choice of the basis and so on was probably covered by it's like when he was talking about the more complicated for for the National Association so you just postulated what Weis the operators the generates too currently but well there is no sense in the models here all these salaries here I have been discovered before us by people working quantum integral system and even sold by by people working at intervals so we can use knowledge from the models here that interval systems
that would get here there is a lot harder and less is known about them but we we have some it would so I was reason and start talking about it the Lagrangian console and regretfully people ignore this piece of supersymmetry Corcoran all sinks into dimensions and wrong because there is lots of things done by supersymmetric community that is really helpful in Maryland right and I'm actually at the nice comment that he said now he understood interoperability better because if it's connected to supersymmetric lock you and he believes that the institution that is not noted legs that he understands interoperability but not everybody knows supersymmetry as much as Vernon so I have to this was a very general things any salary with force of charges has discovered what we're interested much more refined Walker Lagrangian that's is not the the he was here there is 1 Q all right so the it locally the fibers are described by all this thing you have to introduce connection which means to describe how the state changes when I turn on the premiere of that's where cooked so this setup I had here is more like what's called attention OK the 2 days and fiber sentiment but that's OK if you have sole wise In some homework you can turn on a certain number of parameters in this kind of Rensselaer parameter space and for each all I can tell the 1 with break supersymmetry is displays all of the supersymmetric salaries is labeled by coefficients in front of near-term when expanding all the time it was necessary to end the Lagrangian In a the some is also supersymmetric results of it but not what 1 1 the movie I just explained I just explained that you're using same operators like in dungeon likely likened grab your hissing steam operators to forms itself and they operators 1 defines fight work and those forms so the older babies with the formation of supersymmetric story and the fibers are the Fox faces walking spaces in which grammar and studying dependence on so it's like that's 1 of the reasons for example why relations between what to Guatemala were doing and what we will talking here is a little bit well ,comma likes to call the digital equation like generalization of teaching equation but equation has different dimension of base and that and the fiber it's more likely that the Was it is the equation is more like attention OK so let me but actually prepared this election year for myself also with it's almost follows the book said in an aggressive and I were riding and was lost on the drop box at work so In 2 dimensions here I will introduce coordinate their in Lawrence signature but it doesn't matter explores 6 miners With recordings on the cylinder and then I have to applause you don't mind kidnaped last March and undermined his smart commuting variables associated to supercharger and I will introduce shifted course and that's why plus-minus logistics plus minus -minus Keita plus minus the plus minus 1 an old fields in my quantum field theory Will the functions of why this article the Cardinals were perfect there are special super fields which are sold these in a matter of fact the failed they partners are described by victim multiplied which explicitly right just now hearing 2 dimensions structures and no big deal so that amount this a sofa appealed where its lowest company component is given by field burdened with dimensions of each field has 2 components local 0 and 1 thousand times direction in the state's direction there is this is normalization there is a complex collar which article Sigma boxing command which multiplies the demise of the compliance but it's complex conjugate which multiplies the plus 2 minus 4 then there is the family on undercover to variables which are late to apply this formula had structure we had to win it and I would have to look at what future too because they don't clearly spell out words world in books and by no means this explicit formalist people arrived Wednesday and is extended supersymmetry dimensions and actually important calculations for them the apply undermines rail while spinners and then there is an auxiliary terms story right here of solar field which is called age and multiplies highest follower of Quito and this is also the longer they said the kind multiple but described here current will call X and there during my something which is a function of wine plus scores of and firming ones which are called size here the functions of wine why it's a combination for this combination is speak so that the color entity which he will kill field minus sign 1 1 and the result of a a few of the whole and so they to the 2 types of feels gage filled the sky here and here are some scholars but these sales already in In fact multiple series this come and complex this guy is just an artifact of supersymmetry so there's a lot of ones but we have complex color so there is an equivalent way of describing please God which is in terms of what's called twist cairo mounted which is actually a kind of LED lengths at about 1 but special 1 and this guy is I called in sick it's a related to that and the subsequent derivatives the 1st phase I don't have to define its services is which was expected to last and this is the limit to the mind but his scoring because of a superstate so claiming that the visit I has saying information is and when you write a Lagrangian and this is his nice expansion is lower component is given by scholars power expand this thinking then I'm done with this fall so now I will replace the elected multiplicity usually people work with was interested led off and I need them so I want to show that I showing some little tricks at the rear areas using necessary look at this 1 so well what we have here and White will I have to define so there is another 1 which is why it will die the award will be lost money in the long minus -minus the plus-minus to plus-minus 1 so this
is this is the kind of engaged there is no gage filled in this formal only ,comma trenches these guys this Sigma knows that Muneo genetically more correct to describe things in terms of signal because of the potential is too much of the information disguise written in terms of physical degrees of freedom ages an looks a few the game so we have and we have a chance to field land and sea and if you look at the scene might ask him what a difference is obsolete refused to 4 as the 1 is also to form so why don't I just read it to be shipped by but it's not possible because is DEA and the consequences ages arbitrary to form it is a curvature of connection to the plus a square so you cannot do that but then we will I will play some tricks about this definition so for example the and so on so it's important that these guys here and now Lagrangian points on the day of the year with final it was the head 1st time in the history office or stories 1 I just want to age if you want what we are correct I just want to write so there is a conjugate here which is key to topless bar times they 0 plus a 1 thank you OK so now what is the Lagrangian more general Lagrangian of equals to celery in 2 dimensions contracted by 1st remember that gage field we In single mom should be in a joint representation of sigma is a joint representation of gage groups now introduce again I have a good I think our joint representation of Gabriel Michael Sigma it's a sober field miss so it depends on us into functions with space but it's a special functional for perspectives which is of the type here and this is an urgent representation and met field I think some representation of our I call them X found representation which is director of some of its political space-times irreducible an our eyes will be irreducible era of Jake the 1 I gave their presentation debated by the Senate representation there is still some freedom because some lots of parts and Lagrangian no fixed I'm restricted I gave my action is very much fixed now so where is the freedom I assure the freedom and these places are important so under the action is an integral part two-dimensional space which I called exigent explores the miners there is something called the term which is integration over all 48 hours I have 42 I have 48 because I have fought Everything is a function of his office space these forms and the first one that I want to get into Lagrangian is a free New Square I want to go these guys quit because it's a real so I would like it had been used Is it I integrate over 42 Sigma time signal by I have wanted to hear I will get to when multiplied by conjugate all of them will be independent so I will have this time sees itself and there will be some other time so 42 up square of this the correct square the buyer will be deminers to deploy by and this will have a used so here is a thing you blast sober thinks good at another time I can write Tracy in a joint representation so that victories in London the of represent the Lexus last now because any function complex balletic something which later will become Keller potential so let's call it killer potential now it depends on the box as to those the output at which is a matter field he said that the bar I put announcing his coupled with the gage feel they have to write how the matter fields are coupled to gage fields in the correct way is to multiply my explanation of the overtone where reason that the promote led this 1 and you say I cannot write that pair signal because similar campaigns of renew the counting but it contains a thing in order to write the interaction of the matter fields who is With gage fields I have to go out to director malt like and this is this will be supersymmetric for any case now this is not all all this was written in of the full integration of teachers so I have to eat being the barriers in an integral 42 but there are 2 other things possible to write which includes integration of our whole Norfolk part on the initial prospectus although only 2 and the way introduce everything that to possibilities and this the possibilities of course are connected to the existence of Q and Q the 2 different ways that I roads or also to research so plus integral would be to work and now To integration I will write Pete applies if you don't mind so homomorphic on I noted here to deposited minus and the whole Norfolk ones will be considered so there is such integration and I can't seem to agree on the jury function of X like it will be supersymmetry plus complex country because this is a homomorphic function effect doesn't depend on its so this Lagrangian will be complex analysts complex here and this is called and after that was the this is called the Nafta and now you can probably realized that equally I could have chosen a different combination of teachers to my whole Norfolk variables like up last entered the mine is I keep this everything else so that's right the integral duty to apply the demise of his and now we freed them here to write a function homomorphic function of In the last complex country and all of the parameters set your priorities wanted from me are here overturning focal Norfolk function that affect or home-office function that returns of Sigma arbitrary killer potential came and in the 1st 4 games with the 6 because you want to edit manuscripts we know where we at been used its income while all this guys that was arbitrary coefficients so if you wish was arbitrary coefficients I just wrote to structure there is nothing else you can get more general now we have to play with in our job to use the general knowledge we have and see how much we can get around the rules and thermalization group and things like that how much we can say about the structure of salary when we changed managers but what we know that if it's supersymmetry is not broken it always will have that for all the difference will be this function will change the function will change these functions will change and interesting must of our also will change the way it was something else which is a function but we have this freedom of the function and this is a big power allows us to say what think so let me introduce from to 2 other signatories here and then we can make a break and then continue With the solution if I'm too slow please tell me and I if I'm too fast also tell me what you're going to fight for it I'm too slow ascent to the north good OK so then I will introduce what's called the flagrant because this is a quantum field theory there are lots of global symmetries I described only local symmetries the gene was local local gage symmetry know what kind of global symmetries we can have 1st of 4 obviously the global symmetry we can have is already on the blackboard
because in a multiplicity space when we expand the orbiter representation in terms of the proposal representations we haven't space and for each unit of of civil representation and we can have unit Oregon acting at which is can be a global so maximum Global Group G Global it is prodi all of you that's available we just think about Clark right there ,comma lots of them they're there for each quarter of the animals and and I I labels human quirk that in my office and then we can have a rotation group makes exams and this is literary group your mind this can be a maximal look to him of course this is broken clearly the subgroup of Jim which is actually global growth and it can be broken but this this function have to be very under the protection is almost function so we have a lot function which would be warrant the interrogation of this just to figure so frequently that there other representations of mixing the representation so we have to find a real global symmetry group and this will be important in other think that's everybody had to ask me who have taken quantum field theory course because there is something important each gene has you want factor I did not think that GE was a simple grew worse and simple group or something G the group can have some U 1 factor in that 1 factors we can define magnetic charge we can't question you see on the streets in in form of me all the way to make this is the new face of soul the axis will be in accordance so the doors nontrivial faces the risk of getting a signal so that those are the things in the sees environment and energy device Spain were actually might section of and I will get something complicated so for example thank you for your can assume from the beginning you can assume it from the beginning the best way to describe it taking and take the action of the gage reportedly essentially fact In Europe on the factors or is all subclasses off also raised the Tibetan official statements and we will see you could start saying that the axes according taxes according to some probably always will be a lot of money so let's send say this way these are the would the nation two-dimensional engaged the nearest signal up more of these two-dimensional sewers which engage salaries OK it's much so this sector where gage field will call at Cologne branch so few they single-minded under China's no branch and arising where these guys lyrical he Ratso so if this expansion :colon branch are interacting you can describe salary by integrating old-age feel so you now living in a factory Of the space were actually under the action of the group or if the league's branches massive if you can integrate out this massive guys and having fleet expansion of something conical bra most effective actions are impossible to calculate so we will have to figure out something clever way of dealing with that but I wanted to introduce 2 important parameters which will plays a role which will be part of the parameters on the base In the what you want and then we'll make a break so if gays Group G has 1 factor for example in Group G is you 1 itself or product of you once you can introduces an nobody will something called the teacher to engage salary which is something the time integral Paul your two-dimensional surface all of of which it topological turned into a menu 1 factors you can introduced many of them in a labels the 1 factor want to our whatever but this is not good enough for us because it's story has to supersymmetric was forced from charges and turns out that this is a natural supersymmetric generalization and is described by that's the kind of plants are included here and that singles and included here so what would we introduced and has there twisted superficial things at will and will it is a simple way to write that soon would do well there will be 1 power these guys will be cheater I can't wait to do many of and as many as we have you 1 so this is a routine use of that plus 1 and then there is a term in a twisted to 1 particular star Michael W 0 or governing team led calls which is equal to integrate the 2 of you to plus the timeline of cinema To Sigma enabled here will now if you will it's written that said things like which is the a billion 1 would have these guys and then I think place there's which gives me an and was on active here which will look exactly like but I wrote services but I wrote in terms of support everything within 2 hours of supersymmetric so these will be called the judge's or complex if I did that which is this has a name the are is called just to remember for future is called the fire police parameter so complex a fighting she is the sound of 2 plants by hard work are used by the police were so now we introduced we singled out the digital complex survived persnickety W of 6 another question you can ask How do I make the guy's Marcy can I make extras must and answer is yes that was the master is also included here and it's written in this 1 let me single adult because this is tricky things that inimical to supersymmetric salaries but the masses are complex numbers so how can this be complex numbers it's very easy that's the masses of acts complex numbers and not the real numbers called twisted masses and their possible to write 4 global symmetry a generator for any global summit generated you can write to the task the field at and how does it look like indeed the global symmetry rides as massive there already were written in the car potentials there but I will singles them out you take In representation are all following thing the come this is mission conjugate of super filled next explanation of the will college turns the identity In representation are claims 6 so what do I have this representation wherever singly I took identity in adjustable complement and I did some stupid food with will in this company so if I have a global symmetry which is during Ruprecht probable view in my for that 1 I introduces a field VI and VII as a director of multiple itself can start with some confusion and might well he took plastered deminers but so that multiple it was touting his lowest component was this and these number of which but in order to write this I need the global proved to be present for every problem I think explanation here and I get a mosque and these are complex and called and will the Oracle twisted losses and why I call white people ,comma masses isn't because we expanded in so brief you go to the regular refused this will be quadratic chairman next to target time sex with some contrition
social call so I sold that this is a general Lagrangian the general Lagrangian contains possibility to make extra general Lagrangian contains possibility to have this times and which counts among apologized to the world fresh and classes of our dude you bundling has a new 1 factory losses and the everything is on the blackboard now when I come back I have to construct the quantum interoperable systems that I was saying that for a fellow users is a quantum interoperable systems have to describe continent the global system and describes a bundle IQ bundled which was parameters which are in W W will that Inc initially have 10 minute break or something I want to be more the twisted mouth all multiple it's some particular cases of Coke cocktail multiple at just a multiple at which says that it's a function of these variables why not all betrayal but there was 1 and then I specialize in vector wanted twisted 1 and so on what I said but if you take yesterday was aware with us in question ,comma if I said that think the people of invites said the thing whatever extending the law component I called again the signal With the function of the page I don't want to have many gage fields in the car sector so what I say that I single out those those article gage fields and methods and Michael the and you can put on accent fiction you want you can put an extra minute amount principle but I'm not considering I lost my train this is what what I called was a government that is and these multiple it comes in a joint representation once because I want to have 1 the fight Briggs's relations and I think the to become the it's a it's next already II allows also next but I don't allow gage fields will be next this is a nation makes Sigma components expressed in terms of the components in the IIP defined the the time it's a function of signifies allowed I'm sorry I don't make you discuss it after but you can but there is a specific 1 which I called the single and in that 1 I am allowed to put any function and home office by the mirror symmetry replaces 6 by 6 OK so now I want to continue in the mentality of the Wilson modernization group announced is a question that if I yeah America failed to be he called the theory with must get I consider salary with Moscow all are are my master but what I can do I can in passing to grow and this is no general argument which can be which which is made mathematically precise In a little bit of time In a physical description I can Paul massive well last fields came in reaction with something like that .period so the golf center and masses so I degrade them out and what do I do I get this section back but this will disappear because I integrated all of them this will change this terrible distinguish that it had no derivative this will change that's good effect so it wouldn't think this will disappear by teletext this change will have former similar to what was written here and will have the higher powers and so on many derivatives so would wind up wind up with Lagrangian of this 1 there needs to be calculated and this 1 it's the formation of this so if now yes I would use will be in here this rule changes with legs there will be some Lagrangian which will be case of Sigma Sigma bar with some unknown function which has become collective what I'm emphasizing care that this procedure of integrated out massive field goal was the motive in the space provided by and structural Lagrangian steel will be integral due to at the 14th or something which depends consumer segment with number plus integral all due to the future plus if you don't mind as well of something which article doubly effective will die of sick because there are no other fields exercise got what I'm saying that some major police and God calls you and tells you this is an exact integral what Hamlet will be a test event this fall 1 article would be 42 and wanting to go forward to plotted to minors but this is a very important input supersymmetry and home-office are heavily used the family's home off the consignor we assume that the calculation of quantum corrections do not break the home office and 2nd we assume that the can be calculated and in a quadratic bottles this time we have no take so they can be expanded as accorded parts and so on and on and and this this is all why this happens because I assumed must if I assume must go on and on TV alter our he then indicate will the NW effective now statement mathematically justifiable me provable statement its tail and some people would consider this a domino effect 3 is exactly calculable it is my wife and so on but 1 . 7 0 could so what remains determined to get to the heart of the of the signal is not during the presentation of a mail signal is metrics not interfere bill going on here that we came anything I will know his exact answer is 1 thing but I think it is only calculable that I will calculate but the salaries in 2 dimensions should be normalize that's all I am asking all normalizable so which means at the end of quantum field theory with effective action should have been made the I'm asking questions what they can't field so as well with 11 version makes me and ask question so what kind of form can have on any scale or even better easy completely what kind of force and I'm saying if supersymmetry not broke just look if supersymmetry is not broken water think I'm left with is the 1 field figure which lives in a joint representation engaged so what I will write in with an intent of the you stigma and analyze constants of these and that and all these kind of things like so general story told me that I can have without would give which we call to potential twisted affected sets of a potential and we can have Connecticut and higher the would so I collected Connecticut and all hard to achieve terms which are callable in derivative expansion ineffective use of language and I this is all things can happen I cannot completely rule it
signals spot where can I good thing most complex country enhances new has been discovered 1st in old days by did hear that .period Salomon son and the collaborators and Lucia is 1970 for CPN models sold recipient model would be ideal and copies of complex plane yeah a C right now it takes a group JW to so I have it's a mature view 1 and I have be on the basis of the city and this and so that the picks branch will be severe the theory and they calculated effect on a cool branch not and explained when this procedure calls calculating the exact action on Nicole Brown exactly all "quotation mark Bloomberg and as I said it's not doable unless I make all our eyes all eyes must if I make all our eyes math that is what I said make sense if I don't make those nasty like an integrated world that will be points in the parameter space when something will go wrong and something will go wrong exactly point when the masses of some fields integrated out actually happened to become the so I was not allowed to integrate them out massless being so this is a philosophy no 1 important thinking that the punch of all this stuff can be ignored after did topological tours because of the political twist in a case that the political salaries Singh simple mathematical defined this is an index cards using index calculation nothing more In order to get just have to do some index of course even if you can't go far as saying that the let's define doubly effective of signal which is equal to some determinant and then we can argue whether this comes in a physical so but this is calculable and I'm giving answer for what is the would affect stigma for general celery 9 please keep in mind other than I like so I don't really know what do it the leaders of the 2 of you will I would have said here was that the parts integral let's think about the passenger integral and and I want to do this but expression like quantum of and I have some fields in passing the bill which must In perturbations each time I can expect In 1 last given must goes to infinity assistance amenities so infinity massive heavy guys so the couple so that's good to know every book Power won award will multiply due because otherwise I totally did what you expect of them and I that there is no mention of without when the problem among our goal was to go that extra Mike and comical that's what I mean by that would affect the rest of this that I cannot calculate moreover I probably will never be able to calculate the line I introduced scale I said it's in the Hamas so that all extradited all the guys always have mass neither of these monsters or so I have for each of the guys country I can actually even what contribution of the 1st quarter which has nonsense and if it's I take the latest 1 then I will don't care about anything else but that course and so now what is claimed here is that you can or will this complicated when he lost his difficult questions you're going to dereliction of something that discovery something I would discover not asking some questions because all these guys that you're asking about are in this category protection in the this 1 is unique to define this 1 comes from the origin of W effective that I had over plus corrections and I will show you corrections and he will realized that what I will write you already know since 1969 but I will be utilizing in in ?question mark OK Arkansas as give the 1st so that they claim because of the high supersymmetry often equals to although the higher cancer and this is is a good thing right In these objects in the whole of Norfolk and on and on like have to calculate and narrative formula will look at although I will do 1 more thing story this is what is living in nearly algebra all 4 games right so in this not a raise the composition of plus minus and diagonal components so let's take positive note of this so what would would mean that I am sitting somewhere so this is my metrics of signal and sitting somewhere here I glad that these guys in this league also is it is not the reason and were always appreciated connected to this 1 so this is the position I am just so here and here there is a guy here I I've and what they were doing J article could muster only a signal I find his signature the muscle will depend on diagonal components of the same much so as long as signal the difficulty managing ways .period these guys are also must and I can them out all of this is called a billion position I think there encouraged to about that is only 1 little pigs example moreover this part of calculation is extremely trivial and people have said so now I write that affect which only on half time of the age group and that label support our game rupees UN I to gage group to the U.S. just wasn't listed so this is a function of the components of signal and as a strong feeling it's super partners but scenes of this is all office and disappear field capital Cigna's completed the terror by its but only part which is stigma I will be using just more doubly effective all time and I'm giving answers for this and then soup supersymmetry whenever they signaled by supporters of union so that reflect the will signal 2 people start over 2 3 sigma so this is the case if I had you want factories in gage group in this particular case I have only 1 new 1 factor so this is it wanted In general advice many other human factors there will be please trading in representation are which is a representation of reserve of single market now signal believes that the the story masses will leave signal was and killed times the same applies to minus 1 To wrong time since my last call product the algebra and they're always to have some of the positive have signed up for the year so this is the final answer for any representation and if you want to write in the presentation you like you just plug in the representations there's announced no this calculation actually it was only 8 4 dimensional celery was 1st made by Coleman wine so this is a Coleman library computation of the potential is due to expansion that's what I was saying that your nose what happens with supersymmetry in 2 dimensions but it that but the zaniness supersymmetry in this was discovered that during 4 dimensions by and the antelope stage after Oracle's adjudicated .period and so on but we don't care and in this historic parts are important but I probably should be be careful giving credit but definitely in its ankle which was after that didn't hear from them and people so this is the final answer and we will be using it in future from now on now what was a vacuum after call can I
make any statement about what you by way for imports this answer is in any dimension survival in 4 dimensions on the 2 I want to this place out in itself right because I'm using measure presentation this is written for a new president so for example if the seller in 2 dimensions came from two-dimensional hysterical I have culture supplying molds for any old which means that the signal it has to be shifted by codes acquired and I have to have sound there over and in that form and that will make out of that formula classical dialog because if I write signal plus low signal plus I did too it is all the available time for dimensions is traceable contained in the reputation of the codes require more was and I will get some other facilities formal has all the answers for the gold supplying more competitive occasion for the Omega background I have to do some and that's why I'm giving lectures because otherwise it would be much interest so actually using now conclude after having so far I used to be 1 of the most beautiful young of the it's ecological regret thank In a said that for every global growth have every Mendoza global group I can't introduced last I need a global deal to write satiric square canceled shifting his 3rd dimension which is a Kosovo is a global group it's just fine it's enough so divided by Hamas more if the charge is is that it last for the friendship so it's already include if I say that's a color to climb also included here political and masses will be complex masses part of the complex masses for global Group which is a fine 1 and I will integrate them out often because they have lasted as long as it is not equal to the Yukos declined noticeable to hear this is my authority to demand so I read out also integrate out would mean that I have to sound contributions of coal acquiring more and how family so there will be a scholar competition last Tuesday have here for a moment now this and 12 other becomes some complex number of left and right by for every and that is it's clientele which is that the unpleasant doing by and I have to sound is this is itself which is summed up to dialog so if if I have for information salary on total rose I have to do the same thing for that there would be doubled on and blasts and price will come and lost power and prime think double summoned I will get the bomb to that I'm saying is that as long as my four-dimensional stories written as two-dimensional story was infinitely many feel so what infinitely many fields are packaged in this representation with some ants and it takes it out of the arbitrary my formal over there is it's final Of course when you remember that these ends now I'm so many come this becomes very hard on those long form in 4 dimensions of 2 dollars this is very long form but but it's doable let me say saying few what I can use it for you because you're wanted from me that I had to give me engulf water Lagrangian is and this is very complicated and Steven Weinberg in 4 dimensions spends lots of time writings effective actions and pine medicines and so on and I said all difficulty I just put aside I assume that the guy upstairs knows how to set up the little things and I'm interested in this now I make only knowing the I know supersymmetric it can be defined as as supersymmetric work and now the beginning of my talk over goal I have to introduce its operators that I have to describe all this so let's go for years it is the 1 it is already there I interviewed this a master's or integrated single idea I integrated circuit I went and his more it the masses of these guys are seeing my minus images and when I had this lock formula there will be for those instead of and he sued my mind and then I have to sample the root system and that's how I guess this is a contribution I mean look at the school ii losses comes from the the identity of the it's it's something so identities like this is simple and I I don't have any worries that just because it's a sum to the receive my mind images were in use on mortgages on 2 0 2 logs Stewart Cink times log can sum up 2 of the things I wrote for the road time 6 so this is effective action all the Nigerians and agreed on the images and sounds In the end he mind be and wincing goes to submerge thing goes back now With strides we writes about you and then takes place what happens when the money goes to signature and exactly in that place our answer wrong what happens now what the there and I played this kind of game that we know something might go wrong said so that's right watching and see what happens in vacuum what happens at the zoo lack unique they start leaking left vacant in South America and Haverhill these details but let's not because we call the thinking about you and then we discovered that since some paper with chemicals phases of an equal to sorry he did not put twisted masses so he could have found In a large warning limit of Colombia which is a very famous example the region that paper described the Jordan image of Colombia as in terms of getting a signal model in that place for critic for example of global group suddenly coming up to the symmetries and has imaging once global growth comes out to can put his not so the from Colombia we can go to this massive celery and lead to some other branch which is not included in super conformal so but masses so you can leak and this is connected to the question but I want to jump the way it's going right now to speed I'm going great now His legs that I need probably two-and-a-half hour lecture not 3 times but may tempted to afflicted with the Soviets with the book so how do I calculated from this thing of the house like hundreds of Arkansas I'm standing no question of counting the cohomology of these operator Q and what it has to do with the functions that I just rolled doubly effective market good question and they said CPN model is the 1 signal model Mississippi and target is 1 example of the example aggressive money and when I occur but the French representation than that the content and 1 of the best 1 that many many homogeneous cases I can write as so it was a linear space divided by gage symmetry girl but I will give answers for all to get some of their 2 approaches describing vacuum knowing 1 approach is to physical and topological now the topological quantum field theories were introduced long ago problem 1 of the 1st 1 without the assurances paper In the early 80's and the Norwegians paper which made the subject of existing and some so my friend wedding couple of the likes to say that when we introduced the notion of topological quantum field theory he did say that but the people did not fixing their minds the topological quantum field theory is a perfectly physical quantum field theory it's just describes only vacuum sector of the physical so they will tell why should be interesting topological quantum field theory if I'm a physicist and then you tell them you know but his political kind fields not useless once describing such are interesting vacuum structure of physical sorry they say is of course I'm interested in what you structure but why should I stand topological useless and then had to explain that the politically Lasorda's actually captures the vacuum structure so long discussion that he would dilute the VOC physical or topological language because the political stories the same as the Vikings so physical soaring iron mind you was that we had the kind which arose again gage field F 0 1 In gains the
language of the costly space but now I jumped into the arbitrary surface and data places by statements said sea ,comma for the 2 sides Co traders I called in the component 1 or 2 is quantized disorder called money to charge so it can lead to girl I'm not only integrating always engaged field and in 2 dimensions that nobody's degrees of freedom because gage field and has 2 degrees of freedom minus 2 which is because of symmetry to this degree of freedom so basically two-dimensional you left not with nothing but such Fox sold on degrees of freedom now and lots of other matter fields and so on we have to take into account but this is a very important conditions so in passing to we have a sound where have you have to to remember now this is a little bit difficult to work with us in the like that so what I would do and this was invented by many people problem but what I am using his from some old work of mine without loss of and given the Grasso so is that led promote the took independent to form come on that's what we have to do we have to add to the Lagrangian following the stand I post want to wear his number of this you want an integral part of the mother of 5 3 of the 4 now what happens with this is that if I am integrated this over and burst integral "quotation mark create this rule business something another song This is probably what was on some summation formerly of that function all 3 integral trees as far as the "quotation mark integer right because the sound of exponent of an odd times this and this and so on has known contribution only if if placed in the role of traces of a fire is an integer but sporting trades over by closing the door is exactly what I did so these postings are equivalent things that you restrict trees have to magnetic charge and calculates the about integral or in certain positive role this over and and this is the version of depositors so I was inserted like here and I have to say I can forget about that condition because it automatically means there is no way to die here well I call good W. effectively changes doubly effective by ending the extra pair From because a it is the same supra multiplied as Sigma Sigma was lost component of the Super multiple ever for the highest component was Super malt liquor it means that the I have to add the near-term in my Super potential which is toward pies spirit of minus 1 times staff From a equal 1 2 and see In a statement this simple things Mormon and that they have to some over ends I know exact answer for October potential and I can assume that after all is not restricted to those conditions and after that I can now minimize my Super potential because no there is no restriction on the before that was so potential was distinction on the field satisfying the no there is no restrictions on the field and just have to minimize the potential and that's what I will do this is a physical thinking now you answer what happens when you construct out over Soper potential potential so could tension come on laughter you and eliminate obsolete fields potentially labeled with and because I have potential In every for every fixed and have the potential and it is equal to 1 GI Joe trying to applying current minus 1 and I was the W effective with the stigma time to apply a group of minus 1 and I applaud and Jabara plus the affected by this from countless controversy the signal by J and some of the injured when I don't know what the are these guys I would not only if I would calculate Connecticut Energy here but I have calculated I don't know what to do but I assume that there is some and then there is an answer from which I conclude that in every superstar election sector by the settled and which is here the vacuum equations this sequel there because this is absolutely a square of some people Moffat function actually this isn't clear the staff here if I would include in doubly effective I would like this new doubly effective then what and that his new doubly effective Prof modeling and this was a minimum when it's there and that's 0 conditional by a member of the the W priming the decision is that we do this conditionalities 1 by 1 or 2 by of minus 1 development will be effective the signal "quotation mark and so for every any integer said so I take tunnel gage I put in the diagonal content of good group integer numbers I order is an order of tryptophan digit number I have this element and for each element like that there is a vacuum such that the must solve this equation the market like you are enumerated by number of this set of integers and Seamus but I used to describe the effective salary so the question is on what he was doing this and what is in joint representation trades here in our please about this and OK so what would you have to do they have been the example that the example of this is that we must medical writing trails In some representation if you know something presentation how to write the magic from their joint representation to any representation for example if you have a match-fixing in a fundamental presentations the 1 one-half right what is the speed representation you have to to take product what happens when have systematic power 6 metric power with Austrian one-half representation no calculates and then I worry that representation and you take place that for them so you have to look at the Sigma lives in a joint representation please monsters we're not even in the representation somewhere in the global growth so there a direct some of our joint representation plus something written in the presentation and then taken to so when I woke up Texas firmly so that's technical this is just simple way of writing they will end their presentations stories about its future has been on how how to construct in fact it has multiplied many times construct 5 and then take it to a book I will give examples of the use of the let me write this stupid thing for the U.S. OK let's go right for you and this thing this summer no it's it's a correct language that's how they write some more injuries to my mind signature time loss of signal and mind signature plus and minus 1 and I know to colleges that's what but for 1st I want to rewrite now
this equation in a simpler form To just give environment radio for writing single then I moved to topological when so right the following year that for every I have this vacuum and my field story has a disconnected .period so end users who prefer elections factor for instability legs so that there is some over and and I will write so thinks not just stay was because I fixed it OK and I described vacuum that take all facets of and and for each and their old furiously disconnected targets in each said there is a potential minimize potential for fixed and they I have to various that's what it is but now I will end don't but in selecting his formal and take exponent of the both sides I get exponential DW effective this is my equals 1 Simon said all the answer he'd also version of this so I monthly to apply I think exponent of this all of these describes all black so I will travel to the examples will will be little happier home OK now I directed this equation I want to move topological language which is extremely extremely well voluptuous what a topological field revenge for his business this how it was actually discovered 1st the 1st example was discovered in the paper of more aggressive and myself in the nineties which was example which later turned out to be shedding integration so we discovered the sequential to fill so a supersymmetric quantum field theory for which the walking sector was given by innocent croatian so this was the first one and then with for some reason stop thinking about that 10 years later there are small and myself got back to the subject and developed and work with the grass when we started almost everything possible and this was the best of with withstanding the more sinks which impossible and so that we give topological language and topological language What will I will do is legs I had this signal which was from the field and it had gaged the firmly on the side of minor the Anderson 1 and signal if this were the fields In the trooper is to Sigma so I topological so changes the way I think about those guys soaps the blast site mines will be replaced by the land which is now 1 4 the spin no wonder which is 1 4 and the tank which are scholars if you count how many of these guys are that exactly as many as I just wrote because the 1 form in 2 dimensions has 2 components and the 2 others of 4 components and this was 4 components to work for losses before playing this is the same thing to know what is the way it's done that you take us some integral parts in which I don't want have to explain what our cemeteries but there is a new 1 for a symmetry and changing the Dirac operator speed connection which coupled to the magic and tells you what others think of these guys change it by adding the current the connection from that of symmetry 1 and that will change the thanks and the claim is that for a twisted so perfect Sigma it's actually easier the same guys just labels and terms representations the justices of 1 form is a look that was playing 1 of the this guy stays the same it is a scholar archaic that's what you achieved now what about the toys so this is bullet was forced to move to Vermont with what about Colonel multiple what about matters sector and hear lots of things no and they have huge because they told me I used I submit to you want to change this piece of this and transforming to eliminate spinners and I have only 1 forms and colors but once I got out of metals sector Michael group has a bunch of right there this global group is a product of you miso lots of you want I can take a subgroup of you want and change the spin connection coupling to movement so when I write call individual acting on field I will tweak using 1 from from the global and lots of possibilities so twisted metal centers on dumped into dimensions that many many tourists talking so what is the principal that will allow who have acted as the 20 1 of this report charges he this because I want that after with so the 1 of judges to be scholars of 1 of the questions that this is the 1 I called a before you're in terms of the deal principle this fall anything cute is allowed anything because a lot of anything going and it was so there's been connection plus the new 1 coming from any combination of you wants come coming from global growth is allowed within coefficients as a longer that effects which is very important there is a W effects in the origin of celery the becomes after 1 0 4 1 more spot on the signal so any ways which will make the ones uniform is good I will give examples just to make sure that we understand something and this through the main example that we will be working from now on so how do I change Penaloza fields how do I do the twist of spin connection with users the combination to have double facts becoming 1 form which is it's a W it acquires the the index talk about good now obviously what I'm talking about my background match in 2 dimensions Is that what I'm doing and the changing and I'm just changing his company to background matches exam I call it may next it's not the 1st example we discover that this is the 1st Amendment so take representation are to be L fundamental representations of critical gene which I take to be you take a while to fundamental presentations In 1 during so this is my set of representations now ideas masses I call it a a goes from 1 to 10 article this 1 M N and Y it goes on to and I call this my joint there is a must and don't really offer I will take the fact that yet so global growth maximum global growth here jiggled maximum please you kind of you times 1 so you would think that I can turn on Super potential which preserves monsoonal cartons are opposed the use which is you want to the power to help lost 1 all this region supra potential will not so let me give up a potential which is sound Over a B and all of them maybe helped these guys will die as a field leases but then they have indices because there are these industries and here so this I write
them will .period I find was a poet who S. a timescale know what this fight is in a joint fight in any even popped into the power will be also in joint so essays that have to be half encourages and then I have fundamental and fundamentals this is it is clear the fundamentals I join hands fundamental so this is a vibrant under the global group and what is a global growth I claimed that this environment and you want to the power to world-class plus 1 only if masses are related In generally it will not be breaks a completely screwed group what do I know I have but for it now about all this stuff we don't need this anymore representation so the political minus merely a ploy I S a U then they there was the clothes or I as you and joined equals minus you Lin following is a symmetry L goes to explanation of mine and I knew it was you L L will goes to the controversy and 5 close to its financial minus you fight and now it takes these guys this transformations of course we have here you want as a power as we have you want as a powerful here which is different and have extra you 1 here so we get you want was empowered to well plus 1 and for those losses this season OK so now we have to go which is huge you want to power plus 1 and we can use the better tool twists so claimed another 1 it is that in this 1 we can for example do what's called simply sympathetic to which this is infinitely many tourists you can do and I will give a general 1 and this particular example I need for future so I keep it on the blackboard so in the twisted Alexis this fall or is this joint becomes 1 0 4 under too simplistic to it's a 1 0 4 so we call it L & L will become 1 minus S ,comma 0 4 and as half integer is one-half times an integer but this is the integer number and that's what it is and then down real effects we can calculate and what so we have 1 of minors this is what have minus S ,comma 0 4 this will be the forum of the time to and 0 4 and this will be one-half minus 0 4 so what happens 1 have plus one-half is 1 minus minuses to canceled this 1 so this is a 1 0 4 now in general general if we use In this particular example Mauresmo's general toys than this guy is becoming you was 0 for 4 and EU these guys will become one-half -minus as times you'll plus minus ,comma 0 differential and that total think will be 1 of the 4 major and this is true for any annual and so in 2 dimensions this particular example that to media family of course you can do such as that W will be 1 0 4 and now so Feinstein's legs once you twice the salary after a topological which has many many interesting properties 1 important property is its entire policy integral entire president of In an interview integral only or what's called the PSD the previous solutions now let me say what the Pistons notion in more generosity for sympathetic to it said the other 2 is that I woman do so because equations are basically what it says that the entire path integral will have action upstairs will be something and why can't drop everything which is Askew of something and what will remain will be basically is that the function of the equations right now so what do you make 40 exponent of some Lagrange multiplier times quit and I would disagree so entangled becomes integral solution of some people these are people want speak In a sympathetic to for the spectacle of the case actually probably the most general election most generous I will not know more general because it can operate emotional and will specify so we the which is then would you with suspected connection which I have because I keep connection wasn't super twisted superficial times XI where access my fields in the representation of equal mine just eyeballs DW complex conjugate of X complex W it was a home office so this is complex conjugate effects where are all over the a lot of blood this is 1 another equation that Azizi bar the connection so cultural connection plus of X ,comma about equal 0 were new New this is the moment maps GE star associated with sympathetic for ago which is sound i GI Joe Lieberman of sex that's why the extent to which the picture so I have this killer metric due June ball which enters a equation and defines the Jews start moment when new With a simplistic form count calculated from the calorimetry entering in the 1 1 component of the course so there not this system is invariant and direction of the group so that the election of 2 major groups and you going this is used the good situation please cease dimension this model modelers faces final dimension and I have fundamentally too so I started with the interrupt and that was a final dimension 1 because if I'm Dorothy only the correlation functions and this is what political fields oranges correlation function itself purely exact operators which I called Kiraly generators that only I can it is non-trivial tension Lagrangian which are not kids and all it has argued
that they saw no and what is remaining is not only the internal equations solutions of this and this is a very powerful now because some some interesting equations have this for let's go to our many examples in sea waters the equations in our main examples the rules and brother these guys and keyboard joint just let take our main example words that make capital L 2 0 0 and I have only 1 field fired you that can be more simpler than that what is this equation for that part the is this equation that carries his name and it's called the kitchen this right-footed there is no doubt who's stuff because I was I took C L equals 0 I shoulder with these there is no super potential double so this guy's 0 and what is this 1 it's obvious what you choose this moment to map for that think is a commentator of fighting this fight about that shit the algebra .period moment simplistic form defied which defied so I get equation Nabil X X is fine number of 5 equals 0 obviously complex conjugate equations at sea 5 cooks is fight the buyer also is 0 and as the Zimbabwe commentator of Pfizer closed the variety somewhere right here so in the case of people their own the model a space I had a model a spate of solutions to the equation Knobloch is barred by the court there ought to be subtle complex equations to another 1 of these is far less common data of finds the which was it was divided by the action of the game tomorrow and this is called the future model like this is hard work and and lots of things think so 1 doing what we're doing by just stupidity calculating this function where collating ,comma homology on the model a space to solutions of the Haitian accounting for example where calculating portrayal of the manifolds which is solutions all of these questions these equations are written for 31 1 surface Sigma so let's now replace our cylinder I said my dormitory surface of ginger and March .period I had seen in the old ways but the fields are contributing any money calculating intersection salaries of all of these modelers space has been a very old the general this equation 1987 or something and this is a reduction of Eastern tone bullion secretions in 4 dimensions and so on and our probably realize if I can do this I can do that intersections modelers face of instant on global so we are now in the right set of so much general equations of this so this is true 4 . but it is more general and needed that and I recently actually wrote to pay paper of explicit for story what does it mean and I finished today's lecture where there what does it mean but what they said they wrote recently that some whatever there that means and the 1 application Of the value of this model of the equations is falling there has been a standout law very standard since I was a baby the state will since I was little worried man that's a topological story has the following of supersymmetric so reserve following identities take traces of minus 1 to the firmly on the work of explanation of mine is believed to have minus T. all sound over where all eyes are kind of thing now the woman standard Longo's Lexus let's call this of the well because all for every ball on for energy more than 0 there is firmly on the will contribute to the opposite sides and they will cancel so this is reduced only to the change in the vacuum for physical sort of minus 1 to the left in the vacuum gages their Oregon where you will financial minus the arrival of little simplification we go but then there was an amazing which is safe but the story is some finite dimensional quantum mechanics the biological field salaries some fundamental quantum mechanics what is the fundamentals that financial faces which is fundamental and then they use right but this is same trade over formation of minus the ice age some quantum mechanics where child associated to all eyes somehow and this quantum mechanics have been so I'm giving guidance explicitly in this language that this statute is sometimes true in and you some boards that there was I mentioned good-natured than myself in mid-Nineties already example was known for such things we'll talk real clean example was example of two-dimensional young which isn't to say but leads to many interesting stuff and so so I could that this identity can be clearly written In the case of and you which is from the paper I wrote was an aggressive about 6 months ago sure quickly jumped to the same type of financing which will be kind of announcement of the next week's lectures start of next week's lectures is the topological for a of this type is the call all were certain space bees of age to the power G minus 1 of Sydney's being explanation of minus I'm aware of the TI was all I want to consider a bit this is the space of of explanation of the death effect of the Sigma equals 1 this is our equations it with eyes for evacuees and each year has a name it is called the handled going operator and is called explanation minus certain functions which has all of the same as supra potential but into dimension to the couple's to symmetric so this appears only when you have a metric when you have a card space in 2 dimensions for times that there are no idea of how to use of our super potential times 1 on words sigma is a product of positive routes all of it ,comma Sigma which is in the EU and in case just my mind signature and that's it so that's an exact answer of 4 intersections so you plug a function symmetric function on coast of your game sold this equation for doubly effective I had the formula for general case and take it to apology minus-1 1 himself so this is a sum all Jr so I finish today and next week I start with the political staring into to break into global systems will consider Omega background and calculates sinks for a background was 1 epsilon soap-opera next week will discuss something an immigrant and thank you you


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