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Integrability and Supersymmetry
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Title  Integrability and Supersymmetry 
Title of Series  Vadim Knizhnik Memorial Conference 
Number of Parts  12 
Author 
Shatashvili, Samson

Contributors 
Belavin, Alexander (Organization)

License 
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. 
DOI  10.5446/16411 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2013 
Language  English 
Content Metadata
Subject Area  Mathematics 
Abstract  I review the recent developments about the relation between supersymmetric gauge theories and quantum integrable systems. 
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00:07
and and 1st I want to thank everybody for accepting our invitation to come and take part in this event which is very I think the important for me personally for many many years ago I can't promised myself that the day will come and I will try to organize something coexistence and peace said the weddings parents and relatives are here to show a few words just 2 days ability who was a very very close friend of mine I kind of understand best years of my life in the close friendship with the mark this from age 22 rich 26 or something that I remember last time I discussed with him was the the more buying 9 Daniel and the demarcation to some good does work after their holidays in Estonia and my son was born and Jim withdrawal from the hospital to home and this was spread with time and also has many interesting discussions earrings that August of 1987 and so I am happy that eventually this around the place and we can't remember so my talk is mostly I decided after a session Mojica securely onehour goes my job will end up to explain well why certain things he was mentioned I should be true and can be verified that the true so I will kind of build my talk in this year's 1st full of I have to give some technical that's supersymmetry side and then understanding thinker yet so they what I'm talking about is a kind collection of works I wrote to the reason given the Grasso and also we rode paper grossly is that session mentioned and that there was some favorite was Gerasimov and than many other papers before so eventually stand up to subsequent statement I will formally a good doing good way to think are in consider actually include tools superior mill salaries in the postal so in in a sense I basically what we want to draw 1 tool called any higher dimensional in a twodimensional language so for example if stories threedimensional we can't think about it as an equal perennials and coastal was infinitely many I feel safe in a major sector so in general these kind of salaries when you think about this you have a group and you have a representation of this game wears a matter feels which in principle is not irreducible so it's directs sound all sound doesn't want a presentation was a multiplicity is and this is where major field now what when I said that I want to local highdimensional stories into dementia language it means that see for example if the salaries threedimensional but we would want to consider it on Oct 2 times this 1 and that is on this 1 and for expansion the Klein molds that would make me because this multiplicity space infinite dimensional because for every forum or do you ever feel so in that sense the threedimensional salaries twodimensional story was infantrymen inhouse accountants and so on but this is not to the unique way Sokolov declined is not the only way you can think about higher dimensional over twodimensional and in the 2nd I will say what the other ways so as such if there are enough superchargers than any close to inform them into the measures means said an equal to what the Postal means for some of the charges so we made the claim that such salary but with 4 superchargers properly defined means that see the the fields sector losses or there is master of the losses and losses on complex parameters and here they in good situation which I don't want this to explicitly what that means you can integrate out all the massive in 2 directions metals sector which in massive also integrate out None of the alien components but all of the feel secure have after about so I think that this 2nd so if you have a let's say that too you have W plots minus and in this stub their muscles are monsters are related to 0 of complex color Field In this multiple let's call it and then you can integrate out these guys actually differences Marcel differences the way this is called urbanization and I said it's doable in a proper correct way of thinking about saying so they can calculate what you can calculate can only think you can calculate is something what is called effective twisted sober potential which is a function of all 4 of these scholars that I just mentioned so it's a functional food scholars and since I have been analyzer that safe SUN and that's taken or and this will be a function of N and other parameters of course so there is some notion of effective twisted super potential which
07:10
comes wizard for superstore in 2 dimensions was for supercharger sends 1st example of such things was by .period overdue Andalucia media uh not seventies I think it was 1975 or 1976 Wednesday deeds the CPN model so piano model can be written as a gage of celery gaze linear signal model and they calculated the W so this is very old results in principle so this is the exact this can be calculated exactly the reason is because it's 1 of the exact it was not known for fine intent it's 0 1 look exacted says it's along because this is a supersymmetric any close to sufficient so basically what I just said means that if you haven't dimensions force overcharges and in in a kind of would set of which I said That's it would be the major sector is massive and do what I just said this story with comes with this this object which is effective stupid yet so now I want to talk about on the other side and some endurable systems so that community say here also about what you so what you are actually it basically critical points of the process of reportage I remind you said see supra potential is not yet a potentially have to calculate potential from sober potential there is some calculation you have to do so the avuncular are actually labeled by integers and it's an integer said let's say if it was your way and then I will have on 1 end and the integers and the reason for that is that see this integer serve dual use of photos of Fox Association of filtering they they are quantized so I have let's say if I have you and then I have and Fox's legs acting in twodimensional their relates to that in eventually what's important that the equation for lock you arguing by this approach not equal to 0 but to to during principally can't exist no more to the left hand side and linear W so Sizerock immigration there could be more honest ones such as the ones in the Caribbean tourist spots it was my teacher this is gratifying there also in this kind of salaries said in any close to Whitman force overcharges there are other things which are called Kiraly operators is a very special operating if you wish I can that reminded his just for superchargers that's good Q plus Q minus Cuba was bottom line is what you can construct some combination let's Hewitt or at QBE wishes of linear combinations of such that the Hambletonian in the salaries and commentator of Hewitt miscuing dagger or was QB Cuvee dagger cancel percentage algebra says it's Askew square 2 0 Huawei or being worst 0 so as you see it's almost like Aplus operator here region where if you think about she was their deliberations so all our common cohomology also lets say QA if I want to find the ground state agents I equals 0 4 such salary which is this a prospective sellers of IQ zeroenergy then it's the same as if I would calculate cohomology of Q and musical yield harmonic representative which will be the luckiest so if I want to start his avuncular I can remember this question too this operators anyway so would think is it's the expectation value Of these operators can be calculated and this is something something we would be interested so let's call it all let's recall the height of signal sources of some numbers to block you are labeled as a set by the solution to this equation so these are the things that were interested every now and then moved to the endurable system size and what we made the claim was that term these equations which of the vacuum equations that colleges have questions being asked me I know session Bono's but around I can try so this is the that some equations what questions and we claimed it is let's say this way I don't want to make a very strong statement I would say likes that even UN your preferred into durable since then and Europe let's say you want to let's say XXX in 1 Heisenberg Spain chain for example or if you you like 6 0 something I it comes up with some better question tho this is soldiers from better and we claim that there is a supersymmetric come so From the class I described above so as to give group representations of parameters and so on when this equation for that salary will coincide with a better equation for this into global system for which all examples I
14:13
know there is the potential to this equation and solution to the better equation is labeled the game by set of integers where here now and would be related to the number of Fleet's Spain's or or the particle sector number of particles thinking about number of particles that sprints up and down home inexpensively and some not so these equation of that equation can be written into himself rapidity and this has been written lights by people who are experts on this again this fall so claim was that we made at the 2 things coincide with the 1st 1 even says indictable system you like solitary gimmick said zany or xyz or any single I know Alberta model and so on I can find equal also where these equations were doubly effective of Sigma will absolutely coincided with y where identification is basically signified Holtzlander and 2nd is also vacuum expectation values which again are calculated as on the solutions of this equation it will be a for system energies of the idea states now I call agenda because might seem Islam Our and the stared please the Iraqi immigration equation for evacuees it is the better equipped now is wish you can call the experiment observation or whatever whether what you like but for example for the XXX a 1 have on that side is very well known it can give so I have to do a basically representation and identified ,comma sent similar 4 x 6 now these are largest models Simon XXX is the largest model 1 can ask question what about other global system systems so I already the stadium is interesting actually a let make remarks said If you want to consider XX XY model that's against spin 1 onehalf turns out that you have to take threedimensional Sabri and meals all part 2006 and the radius of the says 1 will be related to Zia to the homogeneity here that we move from XXX sexy so in this equation for example a rational 1 this will become trigonometric stupid questions so this is just as as an example I admit focus so now I moved to the other topic which is that you have no income in the hands of the Eastern novel Eugene you go in there if you're restricted to the particle sector and commuting to and from the place of the cattle I forgot to say the problem the cuddling operators forms a commutative ring so if you remember 1 to 1 they said always are the same East after expectation an operator's all eyes will be identified with each sold the commodity Ringo committing come Etonians here is identified with the calorie there of course examples where it's said infinite dimensional like Norton should Inge moderator in particle sector initially the model is something terrible system also which scold missing young system this is a famous 1 was that the function and that his infinitum inhabitants and this can be met today actually that was the 1st example good morning and I discovered in 1996 this 1 should have the chance to do something useful off so this is a very good question so what they're doing is asking in a speech chain I have here something which is the length of his pinching and another 1 which is number reflects all right so I said what was then and this still L enters in at a presentation in digital so if I want to to get C L S I have to take the L fundamental representations of UN on 2 fundamental representations of urine and won a joint have to give them answers for let's say this is a more fundamental justice and fundamental and this is majority and those guys will be related to impurities in each lattice .period the case you would think that it is well in the UK question would question would be how does if salary warriors gage story looks like when representation is costing the bring good question units that it would it's a good question I I talked to some of the cuts will but so what I'm claiming his give me your beloved sister ride my job is to find the gage boring Seoul which means I have to find in which the nation is escapes can afford emotional culture Klein compact if I don't put rose for example find a dimension that I have to find what gage group I have to take and that is adjusting to gage group or other entity that example to 2 the number of flipped think that find and then I have to tell you which representation today and on top of that I have to what the Macias I have to make the masters to the parameters of endurable system believe me every day global system this problem is solved the minister brokers and the 4 largest
20:48
cities in the world with them all the time yeah about the very group which conducted so at the moment is more like on a bet of flight I give you a better equations in Hamilton what's allowable for young boxer equation it to the question How do I change the number of particle right to me or is it the Gates overreacted to gage group is the number of particles soldiering a number of particles is changing the angle gage so if something should happen when it again as he watches the idea what is you couldn't tell a soul older several good questions but I don't want to cover this here and let's say there are some really really good mathematics works done that mean commentary on what social says that not every statement was much much of you know what she was using from our work workers the result of paper when Molleker CocaCola actually construct our from a lot of people with a month of domain walls fingers that it's it's this I want to move to the discussion to the core of the OK so where was the other types yes I wanted to know too kind of freedom I think all in all I wanted to move to the other type of incurable systems I thought that I want to answer his question I'm anxious but in the past me just south of the summer view is that this savory system that I described here is some interoperable system now question is the new ones and I will give you they all come at incumbent on and I will give a better question a question is whether it can recognize a single system on all that said innocuous sectors this salary is integral system now question political quantum Intercable system either I give you but as many commuting ,comma Tony ends as the dimension of the space and I give you all the energy exports from United argued this is useless produced something
23:05
along there was also now I would use of the word is that they want in good this question after the clock that with luck but said this week we can that there are some the systems that they can produce and I don't know what name to give to it and so on but OK so now I want to move to the other type of interoperable
23:32
systems which are called Future Systems head of the homes Nova that did not discover hidden system this way says the point is that say I want to take the Mojica approach that I know nothing I'm just giving you something that you might get excited for clues that might lead to the further that the that said notice I'll have no weapons so that at at the June attended went out with the good at the bodies of pension system so these will be related to the end equal to or superior and males in 4 dimensions Our and this is various things let's say the person who probably formulated generosity was developed and all of them so it's about 2 years ago and it's a way to solve the mind of the story it was definitely to cells and 11 but it's not just that it's OK to do this let's consider 1 surface Sojourners and then punches let's begin Group G go for for my discussion it's you and are most likely will consider told me to be simple story surfers Group G and then we considered gage field 18 and the 1 for 5 so this is an urgent representations and ditch filled with water retains and then he wrote questions which are called fictional creations so we can call College the all this connection Maine and writes this is it qualifies the Pfizer for some reason he called it takes and this is goes on the name of 6 Monday and the 2 other equations it's the is the name of a fight bar equals 0 and the complex conjugate nobles and about a fiveday equal now that here in the room the people who invented instant or so this is actually is a dimension reduction was instant and 4 dimensions With that certain twist when winning the city and the 4 hour called finds the advisory body OK so this the equations and with 2 divided by indications for March so this is a finite dimensional space such final dimensional space there so if it's you told the dimension of this space times the minus bluster for instance OK so now I want a little bit makes situation we are I can consider another space this Medicare would consider another space which will really exist I can cause other set of equations with F of the plus side fight which is now this is complex fight connection and this will be associated with it as soon as you told this is as because these equation is a complex let connection on the same agreement surface and so on and it has complex case transformations I can consider formations to be complex of 5 from the previous 1 side of the divided by complex transformation sketched me this is another now if you look at these equations that lately amid the denies that the day real part of this equation units the equation which you cannot sue real part of this equation is 1st of pitching a question however the Zebari commentary and a regular part of this equation is the 2nd equation plus said the question so what is missing is a right or maybe other we're maybe it's a different standards apply some is missing a blast that space is divided by interrogations formation then this is divided by composition so obviously I can't gage thinks he has to have all unit are engaged in which you so I can't gage figs accomplice direction and then I will get other so the missing equations that In a gage fixing question here so they still space it's isomorphic so bases it's the same space the space love the important thing is that in that picture let's call it picture 1 1st picture agent produced commuting ,comma Tony Adams so functions on that stage yet forceful that space is simply too can actually it is HyperCard about that space is a hybrid kind of fault and basically becomes thus it comes with a city simplistic forms let's call it the Omega of Norfolk Socratic forms so let me take 1st 1 place called easy and Pfizer bar to be these are half of variables there let's call them homomorphic once and visa glorifies it will be untold more and then 1 courageous in Blackduck form data is being waged in devising what trips and this will be a whole lot more flexible electric form on that money because I'm not
30:12
using the complex conjugates of this guy all were I can call home off agreeable they see actually scolded ABC which is a plus I find this 1 I'm not using the substance use borrowed this is completely independent what the complex structure I chose here of course I choose a complex structure because I said What is the What 1 and there is another homomorphic form I can ride which is integral trainees the time same which the 3rd the I'm not using Botts it's just hold more figure and there is a settlement homeoffice stature and disability forms aerosol so obviously this 1 is independent choice all of complex structure on line so this is for a topological this 1 depends on complex structure so we will work now is let's call college I and let's call this chair so we will work in icy Blackduck structure and kitchen cooling they the season in durable system producing the commuting ,comma gone answer now I will take the case of set told will I have that I think has said to see him so taken with family differentials on c my own seek now harmonies Ayarza exactly 3 G minus plus and your and multiplied by Chinese Pfizer bought square now let's see see what it is this is a 0 at home right and this is 1 minus 1 diversion so total this is 1 1 4 and very integrating sick it's all history is the age is coming out because I use Pfizer and defies Mark Watson commutes with Pfizer and I don't use a are exactly as many plus on committing coming audiences half dimension of the space right so he produced integral classical interoperable system examples I II and III emphasizes cause considerable system using Idaho amorphous structure here this as a spaces that space is the same as that state's votes but this time onions are makes sense in this look at examples consider June equals 0 and and punctured so now would ask kitchen so did inventing new indivisible system where this is something we already know that there is this is something we already know and this is called good then think of this particular example and bunches fond of course the positions of punches through related to the coupling constants singled then install a 2nd example stage equals 1 end equals 1 but they engage ruled to be it's so he rolled to Hamadan not only for foresters to watch for you and also sold the city is also known agreeable system this is elliptic college "quotation mark and so on so there is a specified number of punches you specifies a group you specify genius and you get some interoperable system so now I want to know go back to get you out on particularly which disappeared from my then make a statement which is modernization of the statement made of very quickly after divers Britain's war vise and Bostick the gym immunological and morals of Rwanda and Martin Martinique and Warner and the lag in Wheaton and then eventually ended up this for every Page integral Systems there is an equal to or superior meals in foreign dimension where geometry of L'Aquila already we should what's called I would admitted that potential and so on on it is described why the surplus come into rubles is now the identification here are the best way to think about user's forward so if you have a classical algebraic durable system you automatically have something called the preparation let me describe in a very weird quickly what that means you constantly have classical or directed the global system you have a complex 20 ndimensional manifold withhold normal the generators Hall Moffat was 0 4 and its projection to the babies which is seeking a seat we the on commuting ,comma talking with respect to this eclectic form and there is some restriction on the fibers basically fibers from become blitzed alright and 1 can ask how to write the action variables on the basis and the angle variables the fiber so more money if it would be a real case like something which we studied classical mechanics course so this will be a deal to ndimensional any fibers become partisan there is a theory which says that its Taurus fibers of Toro I and we writes the but
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section and the wearables his like it around the 5 boroughs Omega is exact let's write it as a deal someone for OK then we can integrate 1 4 always this cycle in the fiber and get some single Weger power notes on the base because we interviewed felt in the fibers we get something on the base and it's easy to shoulder its ability for will be the BIA which defy purify are that's unified take this away to cast affection invariable knowledge do the complex case so in a complex case as they said fibers of so we have and they have been polarized civilian varieties there's some restrictions there so we have to type of cycles on a fiber cycles and these cycles so we take people need to recover a cycles we get somebody both let's call it a guy and he would take the which is find the words at all were being cycles we get another rebuff let's call it and the cycles of course are normalized and boss a Senator Dole's are calling us here but there are an officer and and analyze them so they must be dependent on interest they have to be related and this simple statement is that there are related with some potential so a outdueled is the equal some homework for their Moffat function face the sole this is against the use of check but so every such Intercable system counters that potential these huge system that I had there has some purple Benzschawel let's call it a future which 1 Irish so we have prep entire showing huge of course is always kind of not very uniquely defined because the there defined after section of some groups because I can have a some missile and I want to explain concepts so we have had for a kitchen intolerable system because fishing terrible system is example of classical direct durable system and we don't In an equal to or superior males which goes on the sideboard went that potential so claim is basically there sect and of course identifications you have to say what with Fitch insisted correspond to which the Missouri and that's supposed story book so now 2 let's construct twodimensional twodimensional effective salary How this fourdimensional superior and wheels I told you that 1 easy way to get twodimensional effective suppression may out of fourdimensional is compact to fly on to hours not dimensional reduced but I'm a compact to fight include all culture Kleinwort's I'm always am and I said it doesn't matter in what the presentation you leave hundreds of weather which presentation you was fundamentally think initially that is the notion of effective is effective twisted sober protection so now but I know said using that Mehsud's it's not straightforward Duchamp identify this relation to improve distillation little but so what we do is different now a member of the Central College and here I want to mention news set time when he was in August 1987 Duma was doing 2 things once singles which become later is and if you remember he had a formal adults notes and so that was once again he was constantly talking about setting and working with said 1 similar sing from the geometric point of view from the representation .period of so this was a good and others think he had a dream to explains that there is a version of the Latin community called Norfolk factorization Zig gage salaries as a factorization intermodal a space of instant so he said he was thinking that if you're having to build model a space of instant answers some cotton unique space there has to be some amorphous structure are and what they were in good salaries system into real mode lets go of homeoffice function of the contract ended ended and he passed away a few months later In midNineties I started thinking about this again and something's happened not exactly which did not want to do that but I should say is it's a president's work does contain some polymorphic factorization is an integral workrule model but it's not home or fertilization waterways because of instant so it's a modular square also the partition function superior and Wilson make background with some simple factor but an integral is a workable model sold the sunset Indian some intuition which at least told me was a question of studying integral saw model a space of recent sentencing I want to say a few words about this so there is a such a mentioned there is this partition function the partition function which can be written as told by such political movement others 11 and so on as it sounds of some single some intervals or model a specific instance on whether he will please explanation of Itau tower is the 1 where coupling constant engage story and a tentative list data so there is some object legs isn't sound of Catoosa
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power and so on and then you have to know what to integrate and depends very much on the fairy and so on but they seem to build their search because model a space of instant on his sentence is noncombat specimen is not momentarily effect computing those integral they so that individuals can be regularized and in order to regularize their and preserving the maximal symmetry and so on you have to read it at the Lawrence Group as of 4 forth which is instituted a nuclear and space this and you can break the Lawrence grow by preserving carry 1 factor here is 1 factor so basically what you do this the season you have to think about are twotime subtle your Honor for and make Cedarville Malay space of isn't lot invariant but they and where for each of billions of rupiah identified crime parameter of financial toll which are complex parameters and that's what did called Omega background because so now doesn't have a Lawrence symmetry you break it by making himself so these former lesser known in principle after you thinks this way you can you can write a formal identified the fields and that's what we were working in middle nineties was Greg Moore and eventually could bank when you completes his salary Weezer classical part in 1 little part that you get the partition function which will depend now on this to excellence and will depend on how long borderline because we integrate ultrasound and other parameters of the now planning when you have to excellence but Mitterrand to epsilon celery actually effectively 0 dimensions at Tsuen Wan Article 2 0 Absolut will not equal to the salary is effectively 0 dimensions but give 1 of excellence equal to their own it's excellent to equal to 0 you still present the super plant Carrier Group III 1 of those let's say bottles sorry here will have a superb ongoing locked in the brakes at 1 it's easy to stroll and on the Swansea and in the form to excellence does work devoted 2 thousand told it did it to show that they are now applying it to this case of excellent people to their up it is to show that we will fit in a model which I had written here so the story will be no twodimensional was infinitely many massive fields so on legal background this is a claim now make background for excellent to equal to 0 is effectively twodimensional superior males from the Class I described in the beginning from the class of let's say the beginning of this stock no equation the remaining 10 we are actually say something about the effect of twisted super potential of this because I said this is in the class of the beginning so the salaries in the beginning and effective twisted supra potentials question is can we say something about that effective I would say 2 things are true immediately without much trouble the first one if I get limit excellent 1 goes to 0 epsilon goes to 0 of excellent time settlement along the as an acute approved this will be professionals for that salary that we're talking about so I have to start with something equal to serve as a set for a Verheugen systems there is something equal to somebody and meals I start with that I put epsilon excellent to have multiplied this the pencil so this is a statement 2nd thing is not it's considerably made absolutely posted hero petulant along the and call the stubbornly low remaining excellent excellent on the concepts 1 and all the energies and out of parliament secondhand so this is some functions and claim means this is effective twisted over potential for that lets say unknown twodimensional superior animals which we get with these alternative way alternative to colored supplying way of producing twodimensional Filori out of fourdimensional now these guys From going previous formula and now specialist at the this guy has a property is said to him that it has a is a residue all the a single Paul Savory regional capital because if you look where described it appears to be like that and then the correction the what if this is of the official that if you multiply had signaled right log called W multiply this by excellent 1 and takes limit you here it is clear this is so now these 1 depends on 1 of Selangor meaning gets right let's multiply by that excellence is doubly effective and take limit at 2 and most here this is the same as limited excellent 1 goes to 0 excellent to era of epsilon times excellent logs it's so this has to be so now assuming it is his expansion in Lawrence serious and the 1st and will be fine it and then will be all
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flatulence and OK now let's now local waters these equations concurrent set up so this a good this is very motivational what I'm saying is of course everything has been ever been checked improved and so on so that really effective PAI will be in this case 1 of absolute and the have the blessed corrections and these follow a formula says has to be told by squared off minus 1 engages sold evacuating the cynical to Fabrianne real that I produced without energy way of doing deducing to 2 dimensions of the articles decline and the lowest on the head 4 the F. B. I the closed growth of minus 1 epsilon times an hour and then the corrections I just took 2 steps Tuesday inside so a good deal of DID calls to buy so this is like Morozova field approximation if we will if we think about quantise in our class rebels right so Haynes said this now we which should think that this remaining epsilon actually has a plan calls right so I let let me say a few words about this so the mother seasonal partition function in lots of 6 are not right so there was a former lover road longer goal was good more for particular cases and then with losses as an integral then there is the formula combinatorial that make to define the term in terms of petitions then there is the there obsess over and that this actually can be summed up and that is that it use of convergence for this and there a lot of whatsoever it's about but this object is well known what I'm saying I'm assuming you know that everything is good are and I want to know utilize a thing this problem so let's not discuss whether that think is good or not it is they said the integral formulas there former Los combinatorial the exact form losses on many many things the projunta to yet so there is a he noted excellence should be considered as age complex if I is that he from from this form 2 W effective the Epsilon a handful should be considered as young young function for quantized pension system is a sad for war this is situations when we have a bet on that and so there is always a potential to better equations sold and then of course the did which himself doubly effective with respect to some proper parameters should be this coddling operators so there isn't time machinery that can 1 right so now we come to the point where the what we wanted to do it is clear to us and now we have to prove that this is correct so the claim now in quantized pitching which means the specter of all commuting oxygen 100 donors .period eyes other given expectation values all falls In what worry I said so excellent to Italy their own excellent 1 equals excellent pain it will total me being for for now I take April and equal to supremacy in 4 days I no explicit procedure how to construct is that really effective as I said we believe that the receivers and then I could only think is amended to check that ended idea and admitted question again are correct quantification of features system which is encouraging and this had been checked explicitly for many examples including sphere with anyone points which is good Torres was 1 mark .period which is believed to color Jerome and then there are these abilities digressions and there are it's clear "quotation mark brigade ropes and many examples alignment explicit chat I mean owners to be honest with explicit check that they seem to roles well I raised the check means that I can always consider the expansion of answers saying that were introduced into global system in terms of this parameter Q which I identified with gage celery making checked and they coincide to a large order my time is up and thank you very much 2 the questions during most of the of rest of the year this seems to have little knowledge of the systems on the use of military bases around the terminal he has not decided are going to integral systems that I want to have a friend unsinkable system than my gage so we will be very very friendly only that were with there are no variance intervals we have to the government has lost his what is it they painted it red and likes to say that now I finally understand why there are resistant global systems because they are basically walking sector of supersymmetric systems and those I understand but the in Maryland lies this is that's what be your statement so when it doesn't understand terrible 1 would
58:09
extensive research 1 and you understand this was the old city then I no use said even I just quoted but I know I'm not allowed to quote person when he makes claim about himself
00:00
Threedimensional space
Complex (psychology)
Group action
Presentation of a group
Multiplication sign
Connectivity (graph theory)
Direction (geometry)
Insertion loss
Parameter (computer programming)
Mereology
Event horizon
Group representation
Lecture/Conference
Subtraction
Field (mathematics)
Monster group
Multiplication
Process (computing)
Spacetime
Theory of relativity
Octahedron
Closed set
Uniqueness quantification
Infinity
Thermal expansion
Twodimensional space
Functional (mathematics)
Measurement
Local Group
Vector potential
Supersymmetry
Arithmetic mean
Computer animation
Graph coloring
Game theory
Local ring
07:08
Musical ensemble
Vacuum
1 (number)
Parameter (computer programming)
Number
Expected value
Frequency
Group representation
Goodness of fit
Lecture/Conference
Term (mathematics)
Operator (mathematics)
Square number
Harmonic analysis
Integer
Social class
Physical system
Process (computing)
Closed set
Forcing (mathematics)
Critical point (thermodynamics)
Mathematical model
Line (geometry)
Vector potential
Calculation
Combinatory logic
Hausdorff dimension
Duality (mathematics)
Chain
Equation
Linearization
Object (grammar)
Cohomology
Resultant
Directed graph
14:11
Threedimensional space
XYModell
Evelyn Pinching
Vacuum
State of matter
1 (number)
Parameter (computer programming)
Mereology
Food energy
Number
Expected value
Group representation
Goodness of fit
Mathematics
Lecture/Conference
Operator (mathematics)
Integer
Length
Units of measurement
Physical system
Compact space
Focus (optics)
Process (computing)
Spacetime
Sigmaalgebra
Moment (mathematics)
Infinity
Mathematical model
Ring (mathematics)
Bilinear form
Time domain
Local Group
Radius
Equivalence relation
Lattice (order)
Angle
Hausdorff dimension
Chain
Equation
Quantum
Right angle
Game theory
Film editing
Fundamental theorem of algebra
Resultant
23:04
Point (geometry)
Complex (psychology)
Finitismus
Transformation (genetics)
Direction (geometry)
Water vapor
Mereology
Dimensional analysis
Group representation
Conjugacy class
Causality
Lecture/Conference
Complex number
Reduction of order
Field (mathematics)
Units of measurement
Physical system
Standard deviation
Spacetime
Military base
Surface
Set (mathematics)
Price index
Variable (mathematics)
Functional (mathematics)
Bilinear form
Local Group
Connected space
Hausdorff dimension
Equation
Right angle
Spacetime
30:12
Axiom of choice
Complex (psychology)
Group action
Existential quantification
INTEGRAL
Differential (mechanical device)
Latin square
Multiplication sign
Direction (geometry)
Sheaf (mathematics)
Kontraktion <Mathematik>
Kopplungskonstante
Group representation
Casting (performing arts)
Conjugacy class
Meeting/Interview
Forest
Square number
Fiber (mathematics)
Position operator
Physical system
Spacetime
Theory of relativity
Generating set of a group
Basis (linear algebra)
Variable (mathematics)
Bilinear form
Functional (mathematics)
Entire function
Angle
Tower
Hausdorff dimension
Right angle
Figurate number
Cycle (graph theory)
Geometry
Classical physics
Point (geometry)
Divisor
Presentation of a group
Variety (linguistics)
Real number
Partition function (statistical mechanics)
Theory
Number
Power (physics)
Frequency
Goodness of fit
Causality
Lecture/Conference
Uniqueness quantification
Projective plane
Algebraic structure
Set (mathematics)
Line (geometry)
Mathematical model
Local Group
Formal power series
Equivalence relation
Voting
Commutator
Object (grammar)
Invariant (mathematics)
Family
43:51
Complex (psychology)
Group action
Multiplication sign
Water vapor
Insertion loss
Parameter (computer programming)
Mereology
Food energy
Expected value
Length of stay
Ultrasound
Physical system
Social class
Logarithm
Spacetime
Thermal expansion
Functional (mathematics)
Bilinear form
Integraltafel
Category of being
Radical (chemistry)
Arithmetic mean
Exterior algebra
Symmetry (physics)
Hausdorff dimension
Order (biology)
Equation
Right angle
Classical physics
Point (geometry)
Divisor
Real number
Partition function (statistical mechanics)
Power (physics)
Frequency
Goodness of fit
Lecture/Conference
Wellformed formula
Term (mathematics)
Operator (mathematics)
Quantification
Field (mathematics)
Absolute value
Combinatorics
Military base
Variance
Mathematical model
Set (mathematics)
Limit (category theory)
Sphere
Approximation
Vector potential
Residual (numerical analysis)
Supersymmetry
Geometric quantization
Object (grammar)
Invariant (mathematics)
Limit of a function
58:08
Computer animation
Meeting/Interview