3/3 Supersymmetric Vacua and Integrability
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Title 
3/3 Supersymmetric Vacua and Integrability

Title of Series  
Part Number 
3

Number of Parts 
3

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CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2015

Language 
English

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Abstract 
"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 wellknown quantum many body systems like elliptic CalogeroMoser system and generalisations. From the gauge theory side one has supersymmetric gauge theories with four (and eight) supercharges in various spacetime dimensions (compactified to twodimensions, or in Omegabackground). 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, YangYang function, Bethe equation, spectral curve, Yangian, quantum affine algebra, quantum elliptic algebra  all acquire meaning in the supersymmetric gauge theory." 29 avril 2015

00:00
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08:50
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26:21
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43:52
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1:01:23
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1:04:10
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00:05
the but we have also and I a man and the we think that it is a pity the the look at the the last lecture on on this topic of ability and supersymmetry and I stopped exactly from a place where I finished last week I should say by the way just cite commented the after talking during the lunch that I said that it's the last couple of important questions left in this topic and after that it's time to move on something L and I predict that some singles interesting will be Stringfield regained but but there was a solid today I will manage hopefully to formally it's a question say I thinks that there are the required to finish the topic In general soap I finished last week with a statement that there is something called Albert with improper potential and there is something called for a potential influence supersymmetric gage in 4 dimensions so this is fourdimensional superior and salary plus matter that was an equal to supersymmetry so there is this notion and there is also a notion of prep adventure both article saying In classical break into system and I said That's the people who names I mentioned last week of found relations between these 2 drinks and I I will start from here so Domingos a vibrant within a potential is completely in parallel 2 of the notion of the twisted effective supra potential in 2 dimensions in a sense that both of them allowed to calculate lowenergy effective action the exact parts that we can calculate and lowenergy effective action but this 1 is for the mentioned this 1 is to dimensions because of the twodimensional again in equal to his salary in 2 dimensions which is a half of supersymmetry of this is just a supercharger this has forced overcharges because of big the nation's this is actually a sober potential and we have discrete lock you out here but given by former water supply and here we call this colors that the 2 were potential depends by Sigma but here the place more sober symmetries and so on that what you are is a manifold solos supersymmetric like you're here are human has some geometry nontrivial geometry and f describes the geometry uh it is a some manifold and let's call and walk In and Out of Tuesday Super potential we can calculate this is part of Lagrangian in 2 dimensions and I said it comes from descent procedure partly it's topological Lagrangian comes from and there is a descent procedure in 4 dimensions also fear which is why engine that is that he is still Lagrangian and of course Balsa restricted without not getting a full effective Lagrangian discovered effective Avandia here we will get all the part without duties so this gives actual potential This is a term in Lagrangian which has no duty which is I remind here just for From the 1st lecturers at the fields this is a of billion hours without further liberalism in this field denoted Sigma Chi and half of AIG sold the stock scholar who firmly on curvature of the gage field everything is a brilliant goes in there I don't want to rankles the gage rule here is a similar thinking denoted by different letters now we have to deliver aid sigh and F again so instead of Sigma unknown will be using letter a this is just what happened historically but many away until there is this ad gives not look effective Lagrangian with to OK so that's a difference here is a potential there is a Lagrangian was there James and that's the knowledge now what about here now I have to describe the main engulfs the providential in classical of reckoned available systems again with keep in mind that there is the function which is homomorphic Indian scholar expectation value areas they say the replacement of Sigmund 4 dimensions so it's a scholar of being the victim of is whole more 1588 opportunities metamorphic but and determines it would it would the Lagrangian claimed was it's a classical of awakened interval system has a potential which describes this so I know when to stop "quotation mark fall to break into the bolsa's so we start with a facetoface which is end and this is a complex dimension complex to ndimensional and with none degenerated inflected form which is cool ,comma 0 it's a whole Moffatt simplistic 4 so we have this the space but this is complex elephants each year just to remember that then we have a whole Norfolk mapped to something which is being which is part of the half dimension complex trade so this has to emotional complex dimensions is half would be ndimensional and then we have some ball inside Syria and and we have this projection given by any function so we need to determine this we need them and functions quite close to each 1 to a change the projection compendium of mud pie so basically simply the the functions on em where is the sum fixed value and a determined that have dimensions of manifold inside the and the condition is that they have to plus some commute was that simple this is not the simplistic structure so there is the notion of Boston bracket which is determined by of these on the Omega and coming this is 2nd this is very rare usual thinking fuel consider every single then you this is real dimension distances that are instead of say this summer's 101 on commute and then there is the seller more fluid Woodward says at sea if the fibers are compact incident that if they have to be taught hearing on break durable system that we have this whole morph everything is homomorphic here so it's important we have this whole Norfolk map but will require that fibers HEP universe of age so
08:50
H is the value of the Hambletonian the Chinese equal with takes a fiber and this has to be polarized ability and and that they I mean this this topics that time giving lectures or has lots of things and some of the things I don't understand myself I'm just using some rotten expert but as far as I understand it this means that they're basically complex so it's again Yukon Pacific Asian holds a cluster the real situation complex plot but that also means that see we have a Keller forum which allows for determine in the 5 worst AT and cycles we have a killer form we have a pairing between cycles and we repeat the cycles to be normalized using Sakala form so then we will have the notion of this thing OK so this is what is given don't usually individual situations which means usually means what we did In the classical mechanics books when we studying in university where told that this construction allows immediately to introduce a what's called action angle variables the variables on the base and the fiber now angle variables will be the angles and distorting the real situation and then the action variables I ii goes from 1 to 10 times given by integrating always is now if it's surreal situation cycles and that is the real toll rises 1 . cycles which I would call a now all PDQ and what has been Q this investigative tools have to fibers the forum on a guy since I was a Lagrangian the the Omega he also kinda receipts to fibers and then this is nothing more but integral all feature which is the 1 for but then we have now it could the exact same thing for is a complex situation and then we realize that sold the summer according on the bays and B so I call it I hear and the corners arrival called Peter in the in the Torah and then simplistic form would be in the real situation of the salmon pink I times the please so the what would you there are some ambiguities of course it's the data certainly love to the exact form and so on but this moment I skip comic situations we have now 2 types of period a period ended the period so I have to define the cool local coordinates on the basis let's say by that formula but I can't define also by this form and I will call these coordinates will coordinates 2 periods and then there is the question that's total all of them I have told an overseas and of this I have to and variables on base so they have to be really I cannot have an ndimensional space to an independent Of course so bizarre and states that son of the ATI Rage DAI underworld and distance from point to an end let's call it a little long ago this year take that has the following meaning also that if I would take the base from there and consider what is that thing then here I could have introduced the let let's say this would be a and that would be coordinates a doll the statement there says that if omega which is a very natural and such linear space To have only going to be TAI which the CIA again it's homeoffice collected form in this space yeah the statement is that HIV aides come from a period over there they're not in this space relation ,comma Gobain issues means said city where sitting on behalf dimensional Lagrangian some among the things that the meetings is that since his guy vanishes sold then we have 1 which is a hybrid will be dying and this 1 for law his exam because a deal that ministries we conclude that the sound all were from to of a dual is locally the exact and defined some objects let's say metamorphic function of they everything curious to find out to exact forms sold there is 1 of the statement is at the weekend and the space of a D & N as P 2 and the which rotates thing candor that is related to the fact that we can go some addition some exact forms with entrepreneurs and Laurel distinct sets moment what I'm interested is that there is some object called so for every and this is reminiscent complex interval systems because of reconfigurable durable assistance under bright because some these are the grumbled because of this and complex because I take complex and then and that's because we have 2 types of periods in there in the real situation we don't have 2 types of .period the and the period suddenly and the the relation defines Lagrangian some money involved this and this is so sometimes the role but also an aggressive and I called the 2nd intervals secondary because there was a 1st in terrible system whereas the fibers where Lagrangian and this is a secondary because this is now on on the double of the babies where Lagrangian someone involved we can deformed Lagrangian some money from some other Lagrangian some wonderful using city a gaining available 100 Diane's but now in this in this system so for example the polynomials Sing Sensormatic paranormal Sophie a or a dual will define some 120 of which I can use To deformed as Lagrangian some manifold and have some 120 and floors and that would be another interval system and that also appears in the business but that's a moment we know we're working with the 1st intervals system OK so now we know that the classical breaking the global system comes with a protection which is also home or fatigue In accordance on inmates so there is now we have this and we have that and his statement was that in the end equal to or superior and may so race which is originally it was by side would return for group
17:36
age group SU tools during the superintendent which means that if you're constitutes a new sorry means a minimal supersymmetric anniversary was it's superchargers then they calculated that these I have written and then the people I mentioned the notice said see that that have corresponds to the prep potential for other breaking the global system of periodic Todd and I have to explain all this incident in more detail so if I take what's called periodic told algebra class cause reckoned the global system it will have staying power potential as I agree with him salary was great group actually after it was banned for SUN group as you and your stupid and mail salary and then there is another interesting example which were already when we were doing this thing which I called an equal to staff salaries which was the furor an equally to summarize plus 1 joined hyper multipleentry wheeze must and that we had before and when I was in the middle that would correspond here and also not to set to any of the cultural mores classical on the right into system and then denied in Whitten wrote some large paper explaining some of the sinks more generality and now it is known is that the infinitely many examples where there is this correspondence so now what I will be during that I will 1st explained a couple of this examples and I will ask the question but that will be the question of the type I was doing before what if I want To take exists classical to break into the mobile systems and quantized consider its controversial what will happen here now I had general approach include dimensions which is important In twodimensional so real that soon the cylinder time Torah I had general approach exists true plus K dimensional salaries viewed as a twodimensional so it was infinitely manacles declined lots had some doubly effective and I can find intolerable system which was saying that we have to do it is a question is very similar now so I feel dimensional is the Times Toros as a sort of assumed infinitely many fields and comes with some effective Super potential which has a property that is critical points are back you up and from the other side exactly slap you corresponded to some quantum interoperable system all States in quantum intolerable system so now I have a similar question I'm told that there is some classical to break into the bowl system and there is quantities should please I'm asking questions these days quantized what they would always a picture here because picture here is classical and although it's a fourdimensional quantum field theory but it's effective action is attended the forecast of geometry so I quantities Glasgow dosages and what happens here and I will give users answer to In the case of these 2 examples in pure of and celery or of the and equal to staff salary of an aggressive and I found very explicit answers and then we made some conjecture sends and development have been so which I will be telling you know talking so let's start with the 1st example so we get some flavor with celebrate into global system signed 1st example for me will be a to colleges was and you will see someone some flavors of my 2nd lectures the beginning of my 2nd lecture because now we're talking about quantum anybody systems this is example of quantum amended voters system I want to mention what system and we already had 1 condominium anybody's system which was an particles on circle to twisted pairwise that function direction so there was 1 particular quantum many would assist them in ways that it should Inge operators this is another quantum anybody system which is much harder than this of course what claim will be knows it will be able to get caught monstrous using C correspondence and discussions so what is this is this is the beginning of an particles let's call coordinates often particles Q 1 2 yuan their own hysterical where is the set circumstances at the time circumference beat and so which means it's Q and Q Y I blessed beta are identified and they interact with pure ways but then she also homered on and H tool is probably won't have 2 want to and plaster potential which is the pairwise and the formula for so let's try this way that's plus potential QI minus Q J & Son Mining tutelage where you fuel where to use a difference of the store it is a work permit sell exist and square some I listened Jerry well I only wrote system then squared times West just function times worse function you kill and Worster function it is a double periodic generalizations of wonder where X square potential sold this summer Over and planning to jurors who 1 that's where I want to think about it and then explained that the differences seemed square ext blast and beaten so I take 1 always seems square she takes by end times .period and some overseas or this can be a viewed as a double son if I remember that 1 signed square itself is a of 101 square so but now I want to think about this wire functioned as 1 overseeing square Over the next class some From an equal 0 to infinity can equals 1 to infinite soaring electrical Katie Q 2 the power came UK of where Q is exponential of minors to be tapped so what they do I have these infinite some actually doubling from to 1 of these I sum up to 1 since cracks and the other 1 I wrote as an infinite series because I can expand this guy In exponential wine is better that's what physicists likes and that's what I want to Alexis and that's where I want to think about it because I will be doing case when she goes to 0 which means the 2 goes to infinity that's what I call open system and in open system is just 1 overseen square potential and this is called a trigonometric that's the way I want to think about where functional market so all of the system is classical order breaking the global system the following fans think about facetoface the home OK
26:26
think about it you To complex right From the beginning with was a complex kills where is the identification was a real period the right because this maintains that he'll might be curtailed but it has also potentially subject of his imaginary .period exactly equal to right Hughes will be complex so this will be 1 of the cavernous entered peace will be at the pageant to it so I go now these 2 Our my peace and that he was committed spread the words they 2nd using complex Hambletonian is homomorphic great you can write this was H 2 IRA H 1 will be some moment the H is that there is a chance 1 so I have a projection To some substance then there is the question of How will effectively I can describe his projection now this is a case of going to college or the which comes with what's called spectral Cox so it comes actually was a lots of her and not all of them come there is no water here about lots but this 1 has a lot of character which looks like this we can see the metrics valued function which depends on the scene and our peace and accused that is that the bomb zany and I'm right now in Maddox With Maddox element 5 J equal our momentum .period data Jr plus and by the same Ermita and which is times staff functioned shifted by QI minus Q J 100 normalization that above 0 divide by cater not shifted by and Keaton a word teacher is times 1 mind that J where TO is just so that the function future sound from 5 1 to 2 let's call it catered to cater to the integer plus onehalf mind you wanted to .period work a cute as a power case square who was shot in the 6th and killed in the explosion fueled towel and towels that we have in our problem is I Peter divided by the pact so if is I'd meet a city of Ibadan U.S. Qschool becomes exponential of minus 2 major that will enter here and this is a match and then there is a statement let them coming poignancy all of our problems those Hambletonian said I already gave 2nd 1 I gave H 2 I have to give you all age all ages and I will give it now determined by spectral power according to age which he sits inside the claims star and definition of spectral Carver's determinant of X minus flyovers the a people killed in the closing for invites pansies then each power over X will have some confusion which we did was we determine intensified and the claim is the 2nd 1 will be this 1 and the all the other higher 1 this determinant will have maximal ends power so we will have Hambletonian switch will all be described In terms of the waste function for example comes from some derivatives out of this said the function and so on and so that's basically the fish that that'll come with you you know what he said Will you each I'm expanding during the years of the new experiment because that you have to take sued the it's a double expansion era using Oliver and his fans but here the coefficient confusions of extra some power and zeal to some power I'm sorry I I could ask the they determined that were that's missing we just promises square this will take so much time for me now and that means sorry and so they want to claim I wanted to make is it's at once you write this determinant and expand you have expansion and date there he did circuit fish inside and sorted on right "quotation mark there is a sense of place In the past on the this this shows is this 1 I need that's a new 1 that circuit so I'm writing a note that this is not to say market will the taxes in Florida I want to make 1 more playing here this is some work of creature from 1979 or something I just don't want to watch so now b we had this fight we had staged in worse off and in this particular case they are seeing which is a complex 1 complex Gordon which describes center of mass Of there are obviously Center Central Mosque and be decoupled here because it's a pairwise interaction you can know that interim Masters 3 times Jacoby on all of this the competition of use of car I call it the this is known ,comma comply Tennessee Jacoby and all this scarring after this the hour period that would define the entrance of integral PDQ always a or B cycles these guys become periods calculated on this car was a foreign exit easy now the Jacoby and we will come back to bite is a cop so this out alright we have a and B periods and they will be called a and they title so this is a case when we have a car this is Ros this book with similar interests so this is a joke this is a joke the of the compacted find power and we have cordon from the cover eccentricity related by these formal In the case energies are related by copulating a and B periods on on the field this is only knowledge a need and then you can test said see uh this gives us some some potential so a periods and the periods foodrelated by same formal or that I had of you With this
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statement is the saying that still isn't clear at his office whose inspector AI so that's the same statement what's written here and you can check said the dispute superior now very old Victoria is a lean meat often at a college or a mother so once I described 1 situation I can describe another situation and that made so we just remember that skewers exponential minus to veto and that limit of produce totals over on an old is we take beta goes to infinity which is same as Cuba 0 and we take them to infinity but keep it under to the power to and which is same as to power to entire Kewell final and then you get the system now arrived at another 100 and 12 and particles here will become efficient quit this 1 which came from and always there the following Sunday From 1 2 and minus 1 explanation of QI minus Q minus 1 plus Alaska which is a Q and minus Q 1 and that's why it's called periodic told us that if you just some of these and leave this term out this is open told and in that case you can see that we could be scale every Q and removal underscore from here but since we have this last term than lander will appear here so you cannot remove completely so you can him sings that underscore stands in front of freedom shift Q by local flounder something difference of queues by local flounder vendorneutral appears here so this guy the open systems would be here so that will be open system and this is a periodic In the last 1 interacts with the first one so that's a periodic store manager with any of his suggestions about what was going on but how do exists limit now this and goes to 0 he goes to infinity and I have 1 a since grandeur sings at how much sorry I can't explain there is another question you can ask him any questions that none of the accused have to be defined dinner with your your questions related to the fact that local authorities with so few II of actually don't have this formula but the few QIA also seeing elliptical juror mother it's something like 1 over and at times beaten which goes to infinity plus Q I hold periodic so there is you have to remove something anything with their many questions can be asked here and this is 1 of the part of things that I'm not expanded on his statement at 1 another a limit and when you it was a technical calculations and you realize that you have to do some shifts and so on but interesting 1 is a disease 1 it is not symmetric under permutations that 1 His somatic under permutation of Q I was huge for use at the it's it's only Nunes neighbor this 1 that 1 was paralyzed officials think something's happened with just formerly is that in itself OK so now I I at least managed to introduce what are these models so if you are studying some classical mechanics walk they give you particles legs is now the period disappears because I send data to infinity so there is no period so it now this again hold more fit but you would think that supposed usury only aligned you have this Hambletonian as you say this is exponential potentially goes like this and they tell you now 2 placed key by deal would be killed and calculated spectrum of deciding the operator of things he would spend lots of time before you figure out what to do and the 1st thing you will realize that you these terrorism or situation is simplified because of this terrorism that the unit let's say to particle case remove the central masses PSquare plus exponentialof Q but the ship and now we can start talking about scattering match so I can start talking about Hugo's to minus Infiniti Q goes to the blast infinity you have a reflection coefficient to penetration you can't you can't start talking about but when you have the other 1 then situation becomes Compaq now there is no notion of his mentor for this and so so what we will be doing from now on we will better put this guy here and consider the souring as a perturbation of the open systems it was a coefficient of expansion under square here will consider the salary as a perturbation of 1 square with Infinity many terms in a potential that operative killed In case of they just 1 there will 1 sensitive to particle it because there is still much Illinois 2 1 away so that that's an exists will want to think of it now statement there are died the attributed to cost the feature of the morals of Marcia calls In Mironov and Martina Warner and then argue with him Was that take periodic told which was during close to calculate AIA thing I do when just said and find S K by calculating the experience that they will get a monster as I would with ample potential in World actually begin with an gave description of the model space exactly this language their old cars they to differential and this had calculated periods of the differentially get so what happens at sea or in the case of periodic although there's our friends found that these car was the same as spectral car of the periodical so and then it was utilized for other thing console will in a 2nd before probably end of the 1st tower I will move to something which is called the Hitchin system which already introduced actually not but not as Intercable system as is a manifold the model a space of solutions to preaching equations and I will use statement which comes from Cairo to more inmates care that there is a construction that thing to global system of teaching type there is some quantum field celery with this many supersymmetry which has a model a space given by a by by the base of sufficient operable system itself at the moment is just 2 examples however to move to the question of compensation and composition will change many things here but I want to keep as much as cool as close as possible to the languages that I had here again I really apologize to certain things that I'm not giving answers legs is immediately because there are so many different subjects of different people developed and sold and my job is to find connection between them and if you dig on each wants and that is entire that goes there and in principle I might forget something to say or think
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like OK so now we go to 26 In order to get condensation and use the previous message that I spend already 5 hours talking I have somehow I need to hook up to twodimensional story because if it's fourdimensional so rare they said it's the mindful if it twodimensional 7 cylinder they vacuum are given by disputes said In for the initial so it's mindful of this OK so what I'd do if I would do With this fourdimensional so let's call it the tried 1 1st time if I would consider Coulter declined reduction "quotation mark declined reduction I already answered this question in my previous lecture I would get something like xyz Vermont I said that the if you are in the dimensions and you go down on 1 circle that you get rejects exam model so the dimensions you woke up 6 salary which has supersymmetric lock you up big XXX the saying always consider the international salary which dimension would reduce total dimensions gives this 1 look at you will find such and then canonical to you then nationally but include all colorful claims then the statement was that story will have supersymmetric popular which is given by spin chain of X X z the said the defiant will 4 dimensions and finds that celery with dimension Andrews the 2 dimensions gives me my whatever salary which had XXX as a rocking sector that initial tally for all mature Torres conduct occasion was all local of landlords would you met twice a month and this is 1 of those so it's not good I should that single internal conducted occasion because it was an equal tool and not an equal for 1 in 4 dimensions will give me any call for into dimensions which I could also take because my an equal to so included an equal to so is the defined by masses and so on so anyway so this is not a good 1 sold outright to ease his Omega background was so will we won Epson degenerate on the ground and this is 1 which will also like to denote legato Times article was 1 excellent and there is a superior and well celery on this 1 and that's a pretty unveiled Soeren hazard for supercharging because this Omega background breaks have of supersymmetry I explained what was that what was decided to epsilon but they said that maybe for physicists a simpler way to think about this is that there are 2 ways some metric which is premature of actual and there is explicit formula for that or you can thinks it's doing that rotation of the are 2 with some angle Epsilon and everything you're considering that currently but now I'd like to Lagrangian which has which is the formation so quite position here will be no defamation he he he planned to forming fourdimensional Lagrangian by breaking go half of supersymmetry and actually also super Poincare symmetry and super .period carrier will be on it here but I don't care what you think I care that my authority has to be super Ireland cylinder whatever happens sellers there we don't care so well I'm breaking lots of stuff here but it will have severe Bonga here and the Lagrangian salary is the St. Paul was the same the simple the formation of the original 1 so now I writing disease was only part of the Lagrangian just for some people in the audience to be I could skip and so this is an email Sturm then there is a the interaction of scholars their scholars of costs so this is going on in individual scholars I have to shifted by using the rector field rotating the Otto here I will it's innocent so this is a tool for I have contraction of 2 reserve active feel they get 1 for my shift :colon you 1 5 which is the 1 formed by the shaved clean they multiplied by its complex will play Stars "quotation mark Star which is taking model a square basically sold that I do is a conjugate 1 with the flight BA Pfizer conflicts last usual term onehalf trade commentator fight conducted filed by ah let's get shifted with is big game contraction of C 1 for the fight bar With Epsilon the final production somewhere else my the bar 0 5 square the Gezira is a couple on wheels their coupling constant and then there a I call does 0 trace traces in the joint representation wager sold this is I removal letter this is abuzz on its part of the Lagrangian and what is the use of extra field which is rotation in part to Epsilon so in text do minus exiting the and coefficient I take some complex number because the think about this Lagrangian that's what I did I had here do you achieve twisted it why contracting Sukhois Witcher of engaged With the Fairfield here and he and I had commentator square of 2 fighters which a shift to do is a contraction of 1 form in excellent and that and other ways to so this is an action so claim he's held I could have done by the way the sinking boats spaces than I would have very similar formula but I had to do the contraction was another vector field so this would be the tool and there will be the 1 which would be in space of x 0 index 1 soldier subservient to Absalon wanted to execute the warrant minus X Monday 0 and I could go for that single cells so I will Trabzon another contraction in addition to this 1 linear combination in here but I think it's 1 equal to 0 he said he also the fact that there was that there should be no overview oversight the you mean I or something it's all included so it's olive Tree dimensions Texas 0 2 4 also I write are to attend are but convention is at sea these are too To take kept here has attacked not this 1 so I break here's a Lawrence cemetery but was a time is I did not so I get salary and then in Caribbean space I take up space dimension and make it the so these are too is
52:41
actually a cylinder and then I mean original set OK OK so Bell decelerating as you see now as a storyteller has a Lorentz invariant what bunkering environment only now I dig the surrender like and I would say I want to cheat the sorriest twodimensional twodimensional not was infinitely many fields here How do I package is infinitely many fields well as the obviously the 1 simple thing would be I take any field flies which depends on 0 1 0 2 0 0 1 baht and easy to part ways Aziz R X 0 plus minus 6 1 or 6 2 plus minus let's suppose my 2nd asked Who is this 1 the 2 Wednesday to bar locally and I write this as an infinite son and Over half of year 1 the 1 by times the to follow and lead to a bottle of and an extraordinary show of minus epsilon the drawing some with square to regularize integral ended this way this guy will have index and so my my Lagrangian now but I didn't always the tossed my Lagrangian will have infinitely many fields 5 and there 0 awards which are Marsalis but the rest of them will be masses and masses will be proportional 2 1 were absent 1 of absolute that's incremental and will go to 0 as it will become must OK but when epsilon is not 0 the masses will be define for although each modes and then this can be verified and so on so now are used here and this is report 1st thing as I said that's the main success of the reasoning that ibid which where will to write exact solution tool lowenergy effective action which I said again with a similar statement as calculating twist effective over potential in 2 dimensions was it also in 4 dimensions like exactly same thinking 2 dimensions in equal to the in for dimension is 1 little pigs up so the an equal to in 4 dimensions is 1 who calculation of determinants plus infinitely many instant on correction so speculative parties 1 because of supersymmetry it's only 1 way and then there is the number with courage now numbered about depression is school explaining his original the instant on paper and so on usually comes was instant on surroundings than or interaction between them and so now seconds incomes inimical to accelerate which is statement of home offices that is a property of home physically so easily on will hold more physically was all over here I said it's homomorphic Canadian everything sold the restrictions because of its an equal toll which leads to the Home Office it is a function so that this color and rhythm of the expectation values it's not allowed to have boys spent on an and use them to contribute but which means the teams that don't don't interact with each incident on an anti instant on don't interact with each other he drastically simplifies servicing because the instant on corrections are now all the all the extent of nor want instant answer no interaction and that means that corrections Momper to would've corrections are integral or model is space of eastern town of some and then the question is on what that something is now most natural thing when you have such a nice places a modular space of instant to asking some sense what is the water is it to be able smaller how big is the space and things like that and these are the questions from intersection so redefined differential forms on the Model a space of instant or once you start intersecting Z and was cycles soaring integrating differential forms of the proper degree and so on and these are all involved in calculations of isn't on corrections are not part of but there could give is that correction is just you take this Lagrangian but Ed fermions sold a calculator conducting determinants from here gave it some now Statement Number One is the I'm going the that there is a twodimensional so on that cylinder it has any "quotation mark tools should perform currency inventory at the house he has some effective with insuperable right precise at any in equal to celebrate has some effective recently into dimensions has some effective twisted supra put the West questions what is an effective twisted supra potential all of these twodimensional story and now I'm using the letter a to describe scholars of a nice colors which in 2 dimensions I used to Kolzig they words coming from 5 so this a 0 lots of 5 under disease Omega background of reduction so what I write notes and result of and myself from some years ago they had to somewhere disappeared but there was here the secondary intolerable system remember I introduced secondary interoperable system which was in the space of a send dolls and I said that the relation between a individual is a half dimensions of manifold which was Lagrangian with respect to the level of security for if I said that there was this thing himself so the skin to global system comes it's was its own 100 audience let's call the coefficients of the Times all this 101 inflows from on involves comes with times insists has mentioned and the complex dimension to and deal dimension and the end real 100 top that end 100 that because those times the forget this is a tall and complex dimensional sympathetic space with a sympathetic DEA wedged in a duel as such it comes with a pool of water in porcelain commuting good 101 let's call them age then there is the Hambletonian for all with respect to each Hambletonian let's called 3 times over those from Canyon fault the and then now is slightly generalized statement the disease so that means that exists prep attention is a function of 2 types of variables AT and AT T because I can for each Lagrangian submanifolds here I can do the 120 inflow and get another Lagrangian someone right and it would be as good as the previous 1 so there are exactly as many times as their age if I go to spectral power they just 4 experts that this statement is it these variables are describing family of spectral not just 1 with family of spectral cars and if I take a entity together it will be the generic that hypertext and expect a lowcarb the
1:01:24
degree if I could teach people to the Iraq said we wouldn't spectral Paris our means having half number of the model but with number 1 for now I make a statement he is on his way to the top of the head of the following year that's more complex so just for many voters the eyes if you have Lagrangian in 4 dimension can always said as I was talking many times because illegal operators were some coefficients and nothing something it will be similar so OK are the eyes which visited the salary as the political future of 1 dimension is exactly the same same as
1:02:11
zebra with and so it has a different potential which is related to the Daschle's ibid with by 120 and fall and secondary intolerable system and this is the result of laws and aggressive myself from 1997 so what I'm saying Ed still robust like were doing in 2 dimensions clean music that story will have the potential again and that the potential is related to the library would impair potential by the Parliament Jacoby a quick where the keys at times of humdinger could question and that means it's as they are defined by another Lagrangian survival which related to this 1 by some legal details stored and that they said in the paper was loss of an aggressive we explained that in 1990 so this paper was called Esiason topological Gates hours I really remember actually what would you say that's basically what we did so that means that there is a reason all of the affected was it's over potential now which is a function of a and the team exactly like there wars the notion of potential as a function of 8 which were cool model over the parameters peaks expectation values are now as many cheeses ace they have different role 87 dynamical teas are October look at and the formal was at media day and they showed that this was a Paul of excellent who goes to infinity excellent told the Times story would limits of excellent to go to wrote to infinity excellent tool times partition function when I "quotation mark here both excellence now when I would hear what steps the lands an official
1:04:11
from this expansion I have to do a similar expansion was the ones right would you will end up with Brazil dimensional salary because of the fight now will have to more indices and they worry that will be numbers right and all of them will be massive except 0 motivated the Madison tho the 0 dimensional and that's 0 dimensional salary with 2 excellence has a very well defined mathematically welldefined partition functions a which is a product of the fertility of times the amount zaire pejorative is basically wobbled determinant of this thing I think there's a separate would've Park is the 1 who up clothing for with him is a classical last 1 Our Zeno wanted to achieve has a forum all son Hewitt was power and let's call it this is the end of Epsilon 1 absolute rule a continent where Alice From an equal Zero to Infinity a solid start to his computers the powers there are and the um ends are integral all were model a space of instant on endings than don't charge the regular somehow was Epsilon said it was something specifically model dependent and the queue he is well exponentialof hideout where town is that 1 of keep the 0 lower to hide plus will coefficients was for that watches for for pilot 0 score but for piety Woodbridge's square for pirates I hear and 5 divided by 0 Is there find a financial that way I wrote actually where I wrote on his work you have to buy a them while I'm getting tired looking at my notes it's over there so this is an addiction can now let me explain through sinks here the intervals or model a space of instant answers will usually David Germain for example the pure equal to salary has just identity here identity integrated it's just awarded and the financial and tourism regularization of more and more aggressive and I introduced in 1995 was that for so long but anyway so this formula for CNN's is determined its model dependent from divide change the model there is a difference in curing integrating here and this away as long as excellence on 0 their final between Casillas goes to 0 they diverge so you don't have to really come back to fight to model a space of instant you just have some recipe they said that the role in in here these wines appear to him cholesterol look division was wearing a detailed study by Microsoft In the 2 cells into an aggressive and often gold also probe into cells and to Selden Street so there is an explicit precise mathematical formula for is that there is a precise mathematical from 4 formula for this which again the model dependence and the claim these it's doubly effective of twodimensional salary all that salary on a cylinder is given by and there is a similar statement which is important now for us to another loan you here you know you don't want to know I put to Logan's and that I was look here and I would concerns in the logo of so there is a real forward finished his discussion there have to when this 1 again as I said each of these terms is precisely defined by the authors I wrote here and there isn't a which went in parallel statement said f of a bull'seye Iberg with which also depends on the other explained is actually similarly need of Epsilon One Times epsilon tool of globalpositioning again the same day that I described here and this was proven by Nicky dying to cells and true while many others succeeded in responses but so this is a known flocked to this is a result that I claimed is true and now the simple conclusions which were relieved for later as very much motivational inspirational or whatever you call it that will be useful is that we look understood Los we don't at this moment urgently need water this is not part of a formalize it wrote was always that was part of a 2 1 is what's in 2nd into the camp or just will come to this formal lesson we conclude that it is equal the exponential of W. effective will pay plus all of excellent through divided by epsilon tool another statement which was seeing said C. W. effective all of AT and T and so on is of a type of effort side but wait and a 18 plus all of epsilon divided by itself so this to state during year excellence go to 0 and books OK so but here epsilon to rose to 0 and we defined W. effective as a leading call in excellent too OK so the rest of corrections are of absolute and you have to prove that actually true that it's only all of this type but this is easy to prove president of yes so residue of the called here and similarly here after recalculate doubly effective now we sent 2nd of some 2 0 and similarly residue of the ball is giving way now on each step we have to prove that it's actually likes this because no 1 told in the beginning that too tired for the legs but the it's very easy to prove that set has a properties of its explanation of 1 of Europe's alone onetime subsequent something which a positive power serious expansion of books and it comes from the statement loves and since it's fourdimensional field salary is proportional to the volume of 4 dimensional space what is awarded fourdimensional space wouldn't fly fall you will have both absolutist nonzero which was somewhere here return William is 101 epsilon times Epsilon tool and approval this is trivial knowledge to this simple actuarial calculations is of course current warning so we're integrating over these you want to use the 1 bodies until these it'll bar integral and now we have to Dozier karate and with respect to rotations Swiss that's when 1 of soul which corresponds to putting care minus epsilon you 1 model minus epsilon tools to model so this is a current work Award and was therefore wish to and what was written here it was this formal of more aggressive myself as a ground water over the incident would a space which itself is written In similarly no recording that's called the linear hyperkinetic quotient so that's a similar formula anyway this is 101 epsilon times when someone told as it's awarded more fodder for so you can just buy this kind of onto fields arguments claim that sees Zeev z has to have a form 1 awarded a onetime settlement and everything here has to have less single so when you some up excellent tool it fully 1 award of someone wanted to define double when you sum up with excellent 1 you get and I think OK so this is now a imports and I make the breaking few minutes why
1:13:55
this is a very important now we go out and look on that form that 1 was told us that's what you are are given as a solution of this equation which we know right here and we get that back you are given by Zyban Wittenberg potential deal ATI equals told I can flow through here score of 1 of the 5 with this and that's what I want to excellent times and last qualifications square so that reflected was used with blood became aware that we get the equation the findings of ocular and we get the equation all this type which and which basically means that the Brett excellent each why are these and the lowest order is the boss of so Boras overflow quantities Asian tensile To be leading given by resentment have potential as a the potential and why boss on the field because I spent some time explaining that these guys is a conjugate variables to singleA variables which where all along so we have the statement is the same as mementos according to OK because Diaz DNA was kind of the WTA was among so what we have here that the statement was a rare just explain his exact words unlawful because this guy was interpreted that exact conjugate moment right ending the expansion image Bob Livingston is a given by but that OK when I come back now I will explain what what how we should think about this so modulation is legs that we could not succeed by doing good Torre compact vacation from 4 dimensions we had a classical Intercable system in 4 dimensions choral conductor fication did not does any single Newell was giving us changed again so we decided to build this excellent thing and I'm still we know that there is a W and by all means use exists where all the results of this sort of result of Nikita the results that we conjecture this result of naked and open growth this is more address on myself and many many other things we come to conclusions that the UAW effective calculated this way food leads to the exact quantities is a friend when excellent is age 1 so now we have to check so this was very inspiring and motivational and now we have to chase and when I come back I will check it for is a case of elliptical general which means same as a periodic target ends and will do it in generality informal it's a question the remitting 10 minutes or something right here now what is the quantification if we forget about Gates so I made a claim 1st thinks it's the I did something with supersymmetric gage salary which was defined the Reza classical Lagrangian here which was defined by instant on integral here so all which was as I said that Standard Poor by an aggressive here then we took 1 excellent only 2 0 here boss of Celesta 0 here the 1 and on to 0 and calculate W and then this story completely feats in the set of soul resided in the 1st lecture so like the :colon set the spectrum is given by the what I have to pull doubly effective to be that and moreover I claims it see it gives me also for any time by but this is a separate discussions that I would not wish 1 now let's see what would be the condensation problems if I go so we have to cited the nation's 1 side is the formation of superior may from another side is a composition of algebraic interoperable system and less formal flight forward contemplation problem that 1 would do with such harmful immune system in the following way we take said 129 in writes that peace His epsilon deal were DQ and then we also replace and square by this shade separately needs to be explained by times and plus so so what we did there is a differential operator Epsilon's girl tools some from my Poles 1 to and the tools the QI square plus and timestamp was excellent some I know too close to to each Westerners function acting on 1 side killed energy highquality tool because their mother Hambletonian says they said hold to Hume and now it's very simple that if I declared In this notations Epsilon to remind minus 5 times each bar them to be 5 times bath times New so this world and the scale of vise what I will get that this will be a square Times New Times New plus 1 so here I will get the new kind of new minus1 1 here I will get my news onehalf and the idea killed by forces have swollen and the new tool so this is an operator minus 1 have you would accuse square Plaza New Times minus 1 somewhere function this is unusual formulation of the question this is a operators at colleges and was enrolled so now what kind of spectrum talking about so I'm formulating our 1st formula to conjecture as to what kind of spectrum this equation so we're looking for a quasiperiodic the Semitic functions of and variables so I Q 1 and Q M wage now these potential 1 worsened square OK here's as 1 word X square Paul Texaco's is so in Q I go to Q J is 1 square tight so I have to specify what took what of behavior I request from wave function at that and I claim that I want to choose Q minus Q J Golf 0 I want to choose a solution which behaves side we had a secure minus Q due to the power news now I still new has a positive real part all or in there were around I can request that there is placement of new 1 minus Newell and system so I can request in the other case 1 minus so I would say this is a regular at 0 so this is a it's 0 and then expanded so as a quasiperiodic symmetric and outside this Chamber I extend my candidacy so I shall niche and let's say 2 particles into particle his I Have interval from 0 to be right I did acquire quasiperiodic which means it see that sea the technicals PCE at 0 time some explanation of cousin momentum and then outside here I extend my period said formal I just then the equation the W the seed money "quotation mark is a statement that the period a period the past by the momentum is in so it's actually pretty the and looking for a solution which ethos 0 behaves says Q 2 a power it is periodical extended outside the periodic and if I have any variables is also cement now that
1:23:40
believes that the proper solution that when described in the following sets the main part of all operator is a Sutherland 1 worsened square protection when always seems clear potential which I will call age the age 0 the age 0 has a it's a so from Taiwan and told and these are hydrogen metric functions and that pigs is 1 that say which has the solution of cute type type of fuel for the power knew this a hybrid proJamaat equation so I know exact answer daynight all put the condition of the this area With the meal .period detail I get the equation that disk was a momentum equals integer then I turn on Q so I turn on a faucet is also called Q the skew were 1st amenable to right now my potential is not 1 of ours square but this 1 seen square plus cute time please easily I write what his warning themselves I want international PCI 1 so this is expressed in terms of side 1 has the same property that its behaves as a Q I minus Q jitters Apollo new when QI goes to Q J but it has different was a moment OK so I calculate quads momentum and again pointed to "quotation mark integer because it out of work what while there is excellent everywhere some integer it's always right inside is always integer I'm Penney or they can beat us emetic function resists constant or so I had to put a period city if I would take that guy OK it was not a .period this is there that this this solution that I'm looking is cause Perrier can this sort of annually and order now the next on the iPad square so I won't say it again put his conviction and now in the queue square level I have to both here and in this guy I had to do was a consistent expansion this has to be expended for potential gets expanded and solution gets expand and ensure I put conditions this and I get some equation because a moment I and claim is that the naked eye and I claim that this is a solution in and what about you fight takes it limits to periodic told a friend that he to other case this potential now it is replaced by some the exponential QI minus Q I minus 1 plus flounder square exponential minus Q 1 and lacking forward this particular situation I did everything was complexes home of equations that describe I just acquired the period to build realize but these 1 has a different this equation has different way to set up the problem because now what I can ask I can consider this on the real lying then I ask you to really a loophole on I'm going to about and if I have haven't particle than I have to take symmetric look into this question has been solved before so in case of elliptical interim or there was no solution before but in this case there was a solution to this problem of periodic told that has a long history the first one probably was good student who wrote a very famous paper including some feel determinants and so on and formulated the problem and so on but did not finish it so it's 70 safety would then in a series Tom Scalia ended going fast here after World Trade anyway said that I will now use in sense and then the 90's but the mold this sinks hotrolled and Lebedev completely finished sent ourselves and see the solution D W deceived Michael's integer where W was given In the formulas that I raised completely coincides with a formal is going strong exact coincides it was conjecture waiting aggressive and myself and proven so we just look is it's a formalist segment was completely proven while Kozlowski and duration so story goes like this 1st think you do is the separation of America which means there is some general so remark correct me if I give this a classical Intercable system you write condom Intercable systems at such that Hambletonian commute instead of fossil bracket you have now ,comma equal to 0 you have some differential operators of and variables which commutes are an observer and then there the storm which says that everything can be expressed in terms of the differential operator in 1 and if you sold the differential operator creation 1 area both in terms of solution of that it might not be differential buys awaits equation when is sold it in terms of that he can writes a general all credit so this the agency said now in this precise so we have wave functions let's labels and with a set of momentum let me call is to under and there and variables writes in integral transform we started colonel which depends on the variable Kewell on our right to Thai X and then in this it's like it is it's a combination of Fourier transform a Somali reckons formation so there is some transformation where the EC so this is lower X now In the expertise we function is a product of single particle they sold in periodic publications fond these people I just mentioned the you know what is this sinking residents claims paper and then there is the question if satisfies ends the period the equation is given such that it separates the wearables what equation cues such right so cute satisfies equation queue of plus bar plus Q of its mind as well is equal the all of X's times pure facts where where you're looking for a solution your FairTest which is entire function of complex variable to its entire and with asymptotic which I don't remember but there is a fixed asymptotic such as that
1:33:24
the PCI will be in town so you have to this great available was a regular measures the Q 1 Q and integrated I don't remember the slowness listen songs can speed but you have to be entire and teaser transfer matrix and there is the it's it has some zeros in some positions which I don't want to say but you're looking for such care now that people were looking good for solutions of along styling and DeLay said the scanning and so on and the 1st question is if there is something Glenn W. something like what I called the Yanks function which defines the spectrum then the defines suspected don't yet this is about the question called about the courage then it has infinitely many zeros which means that a W should be this young potentials should be function of infinitely many because his functions in power so they were looking for a solution and there was a new thing in my understanding that came from a study of this people which completely broke up the eyes which was the answer is similar to what we are doing and that's why we're so that we have the same solution in the end of the trial which is full the relaxing conditions said selects the operator is entire for relax and say that it's a metamorphic rocks assumed that it has appalled and some positions and call those positions now when discussing a different type of solution many solutions by the way is there anything you command solutions any solution you take them multiplied by periodic by age by function thinking this and so this is taking care of bias .period and then for this case said I assume that it's mirror more frequent is appalled that positions call it a lack of Queens what went through discussion that type of stuff it where n is the same number as the number of points then presented solution there are 2 solutions molecule plus and Q minus and inserting X a US was a condition the polls canceled the some well if you find the solutions expertise then put this condition then you will find very often in this space which is formulated here because polls Kansas so claiming this is also a better question explanation of the W D a you close 1 which is explanation of troop III times this but this has been this is explosive exact statement but if you take was building equal 1 identified these AIS was also said See these guys had as the polls forced about photography during that space then their condition that Q is a from proper space and zesty leading to the Elko functions on a realignment someone is equivalent to that this is hard for evidence and this is exactly the same equation is they said when you could then the conjectures guzzles contention approved we just saw the formal were the same they had explicit must magical profile this question is :colon so for periodic taught us is that we get gage celery giving the right answer for it if the color general was system as I told you all in the space which I described here In the entirety of expansion in kills a potential in solution we can gets said right solution solid equation assertion WTI Pelosi wants is a condition requires a moment of being equal to 1 of 2 integer and describes song then obviously it now you make a statement at any time series of supersymmetric real salary which in vacuum sector of which you know classical which in 4 dimensions without any Epsilon's gives a classical model a space to be Model a space of some of the ranking grow face space of some of the braking to global system was saying now and as for then they had to the deforming by 1 epsilon said fourdimensional Sabri and said that but we're considering will lead to proper contestation all this other ranking tunable system so we had to produce so we produced 1 nontrivial example because of innocent people were jumping as saying Well what do who cares about you saying and then would reducing vitamin examples that I want to move on but the statement I think it's clear for every time you will have all of these equation over here there is a mathematical formulation was a quality Asian problem of last fall to break into rebel system that we're fighting and our answer is answer for that Qantas Qantas Asians are infinitely many different ones by way now I just made Sayyed comment that when you put excellence and calculators in for for provision function 1 place is a completely unique this summer over recent months you can do nothing think mean it's precisely defined must mathematically there is no ambiguity there only in different models have different into ground but it's precise defined think but Zia proclivity monster is unambiguous but it would've answered was a determinant of certain Aplus opera product of Laplace operators on the status of Otto times after was resentful so once it acquires the boundary condition at infinity so Britain answered is not OK I had implants to write 1 formula which is interesting for the future so they let me right before I moved to the other models and how do on sizzling line and working on sustaining how does this better equation looks like and how Dublin's look like and this and this formless itself lead to some thinking and so 1st I will write to Purdue would you answer they said it's some of her 2 which heard what you saw was a Dublin there was a formal discipline this Blackwater system with fixed structures the W president is broken in surrendering because I didn't think was so English language has is not that flexible I things saying a word can mean different things and 1 thing other languages as figure would find gender and water Oak my English is not good it's not like English is OK so W is is somewhat Potomac creative plus instant because he was a product of Dublin's logarithm overseas and let me 1st give you W Potomac you buy just writings is equation explanation of the WBA I know if I just put here on bit which you want King so this is equal exponential of pie I hello this is for elliptical Kowloon any I knew I'd wager epsilon times product of areas which we will phenomena 2nd has the meaning of where tennis is a "quotation mark product clove testimony it's an estimate tricks for opened problem for but the same 1 appears to in every other problem for this answer is almost like universal formal this efforts the
1:43:08
purists in the if the colleges most of which it has a nice meaning the same formal appears certain differences some number of this is probably for there were no this is a reflection as metrics of the potential effects on on exponential potential which appears everywhere for this basically says said see if it that W of Part 2 which is satisfies equation pw plateau which is the X is equal Texas argument for 1 1 of them is equally long over ,comma that's in a solution of that thing very handy for his muscles said joint Dorsey and was this which of 2 Lajoie his new for the sake excellence dimension for and dimension so you have to to an extra dimension local more everything is here to divide my excellent usage was OK so what about the montage which once held as a closely held constructs an instant on
1:44:36
W so in the paper with
1:44:41
the grass so we had many years ago description which sent go replaced by new language in the paper wrote to his personal best on fire and most languages I find important so I will 1st gives a statement in the paper which is being aggressive and then for after to wooded with Western indifference so take a W instead of as a function of failing to we were written as an integral complex flavor and it has to be controversial test specified it's a long story of so integrated this huge thing that once highly I will now write what can and Q is given so little killed his explanation of Idaho the big fuel there is a certain national function again depends on a model sold big these polynomial X minus any polynomial plus and plus excellent divided polynomial effect women will affect what steps a where polynomial effects these products with zeros at exactly position 12 K still this guy is a function of she is a polynomial degree Q is a rational were described here the only question is what is kind so can kind is a solution of integral question which basically follows from this public itself in this integral equation makes skies function of little so kind the facts sources questioned integral to the 1 against same contours g 0 minus why times logged 1 minus Q capital Q explanation of mindless kind of wine where Gezira Island and the smell of the liquid would be useful and India 0 is basically a rational functions exhilarating duty which is part of polynomial but the product of economic news the used VX blog explosives Salone Times text so that all these parameters epsilon mn so once they have been engineer currently integration and Associates overtaxed of possible ways take which you go to 1 of professional for pocket so this is Tatum where it comes from is a long story so you sold this equation in interrogation by so indicators lowest order there is nothing then a 1st order you start getting things can only get 3 which is logo of Q times Q and then you however that Taiwanese integral resist kernel of 0 1 the the expansion in Q send your take solution implied in here you get no functional fate because solution here of this integral equations completed determined by 2 objects which is genuine Kewell and Warsaw parameterize in terms of the and numbers AI and that's so you get this as a function of and then you solve the equation explanation of did these and then there is correction here when you put Monfort would you now we have to put government will die effective here so there will be a correction to the and solve forays and you find a decent and announcer now important statement is that each term here is exacting Epsom so everything here is exacting a excellent which is seen as a spark so its exact image far any its expansion in Q 2 which is sad the guy explained Q was assigned since last year it also has a minimum size notes that the statement is but these are this actually Convergence this expression Cuen you can draw where doesn't converge In the complex plane of keeps daily duties this isn't an instant empowerment not the Q was expression of Idomeneo Q was explanation of weird I was a kid I angle plus 1 overdue square so it was 1 preservative OK so now where does the claim is its dancers are exacting a sense that in Ypsilanti Mich was the exact into its expansion but this expansion has images of convergence which is not equal to 0 actually sees Q is a complex for a major you haven't accomplished flames of cables and you're asking What is where does converge and to claim is that now can be described nice waiting where does it converges knowledge and also obviously the contested salaries which has many little accused so this 1 had only 1 coupling constant but you can consider many coupling constant and understand what city give lectures on things but now I want to move on on some things that I had to finish now let's use of this statement that was made by going to Morin and scan that there are still worries what they call no Class S than I sometimes Joel Glickman said this clause in there as classes salaries on their turf um claims at sea before every kitchen incurable system observers of but there is a cage celery which has a vacuum described several written about described by that interoperable system so we started so this what I will be talking now uses certain work when crossover Wheaton which explained that these claims an aggressive and I had them in the language of the cook over an aggressive written which explained and as the language of Golkar Wheaton and Kyodo Oblast origin kyodo written papers and send in favor of an aggressive Roselyn myself so this is probably the 2 sides contending itself wasn't panels and 11 others that this is a claim about the source of clusters that I thought was a region intolerable systems mind you we had remand surface With the gene she and and puncture so we have that connection and principal bundle uh what physicists call gage field and we have a 4 in the algebra all age groups which launched by fire and were rights equation :colon which will vary you With we pick the complex structure on signal the coordinates and we were right to question I always forget it's a plus or minus 1 sign is correct and 1 is not a tenable as of the called 0 and divided by unit variations formation mom interregional trade war I think it's transformations whatever gage group is weak mostly so this is a space we consider and the Hitchin claim that this is a space like that so this is a facetoface actually the word not just complex inflicted structure that this space fares but it's
1:54:32
Altschuler has tried homomorphic structure the independent wants it's called a hybrid Keller it's a for the wonderful than he during the described the projection hiding In 1 of those 3 homomorphic structure so we have now the Home Office structures let's consider amorphous structure that means this is a structure where I will tell you what he's P and what is acute which are complex and then there's the Peabody & Q Bar which are complex conjugate I have to tell which 1 is the 1 in which 1 is another so inside of amorphous structure a z is a hazy and finds it Laura are from Hall Norfolk and a Zebari and the Our County home the governance and stability for all major outages Tullamore more fix should depend on on this it's an integral signal all Delta a zany editor 5 the bar and this is a quorum on surface and this is also types In a homeoffice structure leads to 0 because I have both homomorphic but on the single this is this the bar so this is 1 ,comma 1 for which I integrator were sick there is another homomorphic structure which is called the Javal homeoffice structure which I will need later work and that is the case the seals them I J and K I looking at Kazan my lecture today J pull Wallstreet structure there whole Moffat coordinates are CD I will call it which is a plus I fight 1 thing is clear the worst right and there is a plus I find there is a cocoa conjugate 1 complex conjugate wonders I will be using ,comma 0 simplistic form his into Sigma all 3 the car industry which could have been now here I have to I do not think this sold its integration so we have to use his homomorphic structure this simplistic form on the solutions to the Hitching equations which are or here here we have to he was that homomorphic structure on the solutions of the equations a plus 5 5 people 0 divided by complex creations formations and the service statement already use that space as a space is the same as that space and then there is the Soham about them and explanation was given that this equation has a real and imaginary parts real parties the 1st equation imaginary part is the 2nd equation plus complex conjugate and the missing said equations there is a gage fixing an accomplice the election of this equation this equation has more symmetry but this basis of the same so this simplistic form with that is the same space says that's a balletic form was that equation and now there is a statement which I will make up the was just superrich he managed to find the information which might have been so they said it's survival with 1 side or the other I had to write eyes and mine here and so hitching defined the interoperable system in this whole homeoffice structure so they said as a space this is always and face and the same is here and faced is here and there is the same spaces but now we need to define this projection so we need to give 100 tons and Asian define 101 poignancy good using spectral ,comma so some equation I will now the bill for this year tool because I I want to focus later on in history to win have time for it was alleged that the gage to neutral making said torture and I don't want to specify real structure and style Gabriel to which means that the when we talk about interconnections here Instituto and there is a sense to see that you but that is all he has to give 100 torments and here as incumbent audience in a simple in the form which is very similar to where I was writing traced to the power I in my 1st lecture for simple models include emissions here legs and h I goes from 1 to 3 G minus 3 plus spending where is this is a Vermont surface of sojourn you was and punctures is equal integral always Sigma chicken New I times 3 5 Zebari square the square it's just a little more frequent stories about where new alliances are big users of exactly this many dementia linear space of the family French so this is the 0 form and the others that guide me wiser 1 minus 1 differential so the 0 to differential oneliners 1 font we get 1 1 form integrate or signal this outrage on we is a AAplus on commuting because they depend only on finds the body don't depend on me they depend on behalf of themselves wasn't committee here's a balletic from tells us that AT and 5 Zebari conjugated so finds the bodies like order and a like momentum and ,comma Tony and is a function of half of as possum committing desiring and this gives us a projection so we're done here isn't system now some new comments can see the G opportunity to be you and then consider your surface To be Torres was 1 punch OK then this 1 this thing is that if the color during what they said believe is the result of course can aggressively from 94 2nd ,comma suppose JT is as simple wasn't was there and we consider clothing when surface so here we punches I think this is an particle good day system King is so what I'm saying is that now the distal Coleman said that force specializing surfing number of punches and genus of romance surface and the gage group you can get different well known interim systems and 1 was at the college are model which was Torres was on puncture the other was a good day to be assessed which is this is horizontal and the secularism selling generalization of OK so from the other side this sorry now I would explained his Penny equal will start slippery and celery informed dimension and Quantas nation here quantities nation it was a very I'm real sorry on Ottawa trying to Otto was absent so this 1 really that when this way for this 1 corresponds the way I just wrote here it's corresponds to if the end punctures nn people to watch and see and
2:04:26
equals to 0 and superior and real soaring conform to match what I wrote here its flavor group a list of civilians and grant minister calls so this is a summary where we have you'll end gage rules we a flavored group of all 4 of you when which is 2 times and 6 so we have a right of remember I had example I had failed fundamentalist elements of fundamentals and 1 joined in 2 dimensions was my main example now what happened and the terms of the this is equal to about 2 times in similar situation I have these fundamental plus fundamentally Cyprus so I have a high promote fled as many as 2 times college think this is the beginning the sober and on 4 dimensions now what about the statement that 1 might be interested in is what waters doubly effective as to what is our calculation engaged solely in the language of teaching system in general so we concentrate now pretty general station would be if we understand 1st this and this and this example really understand renders this example All Souls and then it is well known that you can hook up indigenous surface by pasting by growing procedures from the sphere was for punches and Torres was 1 punch they understand this too little understanding pitching system I have almost no time so I make a claim the question is what is decelerating population which is the same as what is doubly affected which is same as younger function in the language In the geometry of the ancient wanderlust organizers goes like this the Japanese 1988 and this is the youngest Sydney organizers his brother lost his vision again now it sounds like a good bet that young Indiana wrote famous paper together where Tamil dynamic that Kansas was introduced after this formula very similar to that of a dynamic but that's if someone brings OK yeah and function by definition is a function that better equations are equations critical points of that function note every equation comes from a potential by minimization but this better equations do then it's unknown what take In this procedures upon describing tells you if it's endurable system it has to be connected to gage so engaged so reserve is a potential and avuncular described by critical points of that has to be always say I can't function there is only 1 known example when there is no younger and function this isn't available systems they described for an equal forcibly and males which is something terrible system which of the precise what it now probably has a definition but but that figure does not have the Andean and I don't know why this is a message exists as 1 of the big questions why intolerable systems late with their don't describe it in dance of an end system was in the U.S. as they give you they don't tell you for what it is an answer except they tell its calculates animal dimension but they don't give you a system something explicitly which is that they don't give you this way OK if they would give it this way is on the ground and business they don't give in the ways that we got used to they give you some other way so they invented their own plan for which they have all solutions they need and from the 2nd should have the same plan OK so what window of in order to explain the little bridge over the sink is let's fix some reference complex structure all Sigma well this will be true foreigners and fixes group to me I will now consider president was seeing because this was centrist fled connection so accurate ironies this anyway so let's group to return to relay England adult to ascend to see how this is kind of crazy because linguist to present to Espearson up but in general the physical here I will need to start with the Langlands dual group but with links and that the deformed reference complex structure which I would note on the W W Bar whichever differential menu which is 0 no doubt word W W the body of the W so this is my visit from its own and our claim 3rd term any connection saying component written let's see this is a deformed now this is the same as a but they Omega Barberton deformed complex structure was differential you can return in the case of a central again there is a minus 100 million 0 minus from the Tour nearly onehalf of the and a W which we don't have formed with with a former direction of Samuel and not Neubauer if there is 0 1 2 0 so this is a parameterization which party only the creations sorry this is for fled connection they hear NEC OK so we takes space now claim is homomorphic now to see what happened here in J. Hall Norfolk structure massing depends Blackduck formed and not depend on the choice of the complex structural Sigma it was enough I hold Moffat structure where simplistic form depended on tourism complex structure but in J. did not weigh idea here is that the I'd fix a complex structure and make changes and now I may claimed that following his throat the Tullamore fixed structure the flatness condition the conditions of F N A is a complex of connection I don't use it anymore so I raised its this EIS Aplus scifi from that book for because 0 now introduced in this cage I'd I made some choice I used against from complications from mission to reduce to it is reduced to equation Dubois minus new D plus minus the people's minus onehalf PQ meal have somewhat of a twodimensional gravity political studied this was only right hand side and this was equal to 0 was a constant curvature conditions so now we have the whole Norfolk in whole or in the form of a structure which is June we had the simplistic form which was traced to a wedged which we had to divide by compositions from formation now this is equal or Sigma June ,comma notices this is call their coming wedged the what happened to buy picked sings a complex structure and deforming it was I actually took something which was completely defined in terms of topological data there was no way a single word here what complex
2:14:20
lecturer at peak but after kicked in the form that I will for in terms of their Tamudo entity which actually depends on the streets of complex structure because when times are attending the basis of the original complex structure reference complex structure like this but this formal is extremely important because the for generally months surface actually to definition of what's called a center to fall prey as gentle interloper is the 2nd order differential operator that on minus onehalf differentials widen minus onehalf let's not at differentials Over the mine is the square plus key because this case which I put it If you look on the Woodies remaining after this this
2:15:20
is reminiscent of delivered cycle reduction what is left after the gage transformations when I write this equation there is something left here and they are under did due from office under which this transforms as a tool ,comma 0 projective connection so the tough like Watson in the OK so this is a over and now let's see what happens resistance to offer said 30 as if to operate turns out to be Lagrangian some wonderful he now homeoffice sympathetic manifold because it said would described terms of the conjugate variable is real OK so as soon until all perceived as a Lagrangian submanifolds motorways space of complex with connections more delegations from France end if we have sphere and that was the problem I was having sphere was and what points In lot points they are allowed to have regular singularities which reduces hostile that Keating is given it is the sum well I consider and marked points sound from April 1 to 10 let's call it the idea that he was and what is it you 2 1 1 Wednesday and this formal tells us don't have a suffix and when the plugin and here we get this is equal the sum all the danced is so I've found that I will be the 1st to visit DOC firstorder polling in March .period is conjugates tools a position of some of this is a formal new defines complex factors this is acquired the other article of a projected the words this too ,comma the differential transforming was thrust into witches and that's a simple act so now From the other side what was written here was written completely and topological language which means but I did find good according notes in the space where space of all actually the Lagrangian some wonderful when they change the complex structure when I change is a soap opera will be this language Zia equals constant fire made the constant I'm sitting law graduates wonderful and what the verandas some wonderful this space of such cases right there and apologize in terms of Phase I changes the I'm off in this space I move this Lagrangian someone no playing the device .period is the Haitian intolerable system as originally defined and studies in spectral problems effective from which means so that all solutions of the equation HIP signed people's EI where age now are replaced from classical hammered tolerance by differential operators in some space so this noncompetitive and committing differential which is a condensation problems without restricting too tired of talking with space we're looking no to just just say spectral problem where asking this makes sense and don't have to say which space his statement about what Energis lot probably right in the gym so studying Arizona and 1 of our age minus the claim is there are 2 basic themes which are standing here and this for Group G which in this case is to Will met into the Lagrangian submanifolds a equals constant for me L G so this was necessary somewhere yet this is a group LG for you study Specter problem for G for Hitchens's them
2:20:27
and they claim is there is a kind of canonical man from the space of Lopez the of from this space of the solution to this equation to space of all births in a doll system L S and to see Our LG which is given by Lagrangian submanifolds a was constant In this homomorphic structure so we formulates a problem of conversation in I'm homomorphic structure and explained that that conversation problem is a statement about studying space of stillbirths which is oranges of wonderful new doll system of forested see image a homeoffice structure but energy homomorphic structure this is a classical opera it's a dramatically stated problem in the Ivel Moffat structure it's a Qantas Vacations a quantum no How do we described space of like ways with the statement belongs to general G and and its mainland cementing fed and force the Lord Jesus like 0 1 soaring 0 or 1 1 I already said Hu who who was a statement this was a good then system and this was elliptical Jerome and that then and elliptical Jerome there was good thrillers cleaning and I mentioned people this is separation variables lists this operators in 1 variable this is the 1 variable operator or general problem using many variables but the number of variables is hidden in the in the position in the home and impulsive so this is it this is unknowable the Boxster questions physical 0 1 side is unknowable Boxer questioned sometimes and it is a differential equations sometimes difference equation actually reason is a difference equation for elliptical periodic told us is because periodic told there is a limit of elliptical Jerome was for which this is a differential equation on a Taurus was 1 month .period retained replaced by double periodic function because it's on the anyway so any statements now is that you can find so now we have to describe but relations between and disease because you have to find the comforts Lagrangian manifold for every season has to be so we're looking description of this Lagrangian some manifold announces legs this if you introduce topological coordinates a topological coordinates let's write eat some of our data on Friday which tend have better any topological corner doesn't matter this year then there really can writes a graph overseas Blood ranges of my generating function for the Lagrangian submanifolds as a function of let's say variables on fire and variables such said the WDR and fly is beta and the W the Zinni is this will learn the classical mechanics you have a space the political space let's say the Indian space with some in accordance alphabet you have some kind of Lagrangian some manifold there and in order to describes it ,comma graduates of my fault you have to give basically in the description of the survey according In terms of the political could and the that's called democracy have to find him but then major general definition watchers think it's fair to find such Dublin for accordance alphabet it doesn't matter what coordinates introduced His comments there is some Dublin and I have no time to describe waters according that's for which these generating function is a governor effective a of the administration so naked generously and I found the such coordinates for the case of a city tool agencies such said generating function of Lagrangian some wonderful the purse is given by answer so silly but by way our composition methods in In was described before so I probably have to finish I think you there is no apparent few and