2/3 Supersymmetric Vacua and Integrability

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2/3 Supersymmetric Vacua and Integrability
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"I review the relationship between supersymmetric gauge theories and quantum integrable systems. From the quantum integrability side this relation includes various spin chains, as well as many well-known quantum many body systems like elliptic Calogero-Moser system and generalisations. From the gauge theory side one has supersymmetric gauge theories with four (and eight) supercharges in various space-time dimensions (compactified to two-dimensions, or in Omega-background). Gauge theory perspective provides the exact energy spectrum of corresponding quantum integrable system. Key elements, usually appearing in the topic of quantum integrability, such as Baxter equation, Yang-Yang function, Bethe equation, spectral curve, Yangian, quantum affine algebra, quantum elliptic algebra - all acquire meaning in the supersymmetric gauge theory." 22 avril 2015
Point (geometry) Vacuum Group action Presentation of a group Level of measurement Sequel Insertion loss Parameter (computer programming) Theory Grothendieck topology Product (business) Power (physics) Group representation Mathematics Kritischer Punkt <Mathematik> Square number Diagram Hadamard, Jacques Nichtlineares Gleichungssystem Analytic continuation Length Determinant Theory of relativity Process (computing) Equivalence relation Chain Fundamental theorem of algebra
Point (geometry) Computer programming Group action Presentation of a group Multiplication sign Sheaf (mathematics) Valuation (algebra) Kontraktion <Mathematik> Event horizon Graph coloring Product (business) Power (physics) Group representation Exclusive or Centralizer and normalizer Mathematics Positional notation Thermodynamisches System Natural number Complex number Operator (mathematics) Ideal (ethics) Integer Nichtlineares Gleichungssystem Körper <Algebra> Determinant Descriptive statistics Physical system Theory of relativity Rational number Sigma-algebra Generating set of a group Commutator Infinity Coordinate system Two-dimensional space Line (geometry) Price index Flow separation Equivalence relation Vector potential Calculation Combinatory logic Order (biology) Phase transition Chain Resultant Maß <Mathematik>
Point (geometry) Vacuum Polynomial Computer programming Group action Functional (mathematics) Presentation of a group Correspondence (mathematics) Quantum field theory Food energy Rule of inference Dimensional analysis Element (mathematics) Power (physics) Group representation Goodness of fit Mathematics Kritischer Punkt <Mathematik> Many-sorted logic Term (mathematics) Different (Kate Ryan album) Complex number Operator (mathematics) Cuboid Nichtlineares Gleichungssystem Körper <Algebra> Determinant Condition number Physical system Multiplication Graph (mathematics) Moment (mathematics) Model theory Thermal expansion Lattice (order) Vector potential Orbit Field extension Network topology Phase transition Chain Quantum Game theory Spectrum (functional analysis) Resultant Local ring Directed graph
Logical constant Presentation of a group Group action State of matter Multiplication sign Correspondence (mathematics) 40 (number) Materialization (paranormal) Sheaf (mathematics) Insertion loss Water vapor Mereology Food energy Dimensional analysis Grothendieck topology Group representation Mathematics Many-sorted logic Square number Körper <Algebra> Determinant Monster group Position operator Descriptive statistics Physical system Area Potenz <Mathematik> Sigma-algebra Sampling (statistics) Physicalism Price index Two-dimensional space Measurement Orbit Category of being Cylinder (geometry) Vector space Symmetry (physics) Quantum mechanics Quantum Right angle Summierbarkeit Metric system Sinc function Directed graph Spacetime Point (geometry) Polynomial Functional (mathematics) Sequel Transformation (genetics) Similarity (geometry) Routing Partition function (statistical mechanics) Quantum field theory Student's t-test Graph coloring Power (physics) Product (business) Element (mathematics) Wave packet Frequency Cross-correlation Degrees of freedom (physics and chemistry) Well-formed formula Term (mathematics) Operator (mathematics) Boundary value problem Nichtlineares Gleichungssystem Lie group Associative property Multiplication Surface Content (media) Basis <Mathematik> Total S.A. Sphere Supersymmetry Invariant (mathematics) Network topology Calculation Object (grammar) Local ring Maß <Mathematik> Gradient descent
Group action Presentation of a group INTEGRAL Multiplication sign Direction (geometry) Sheaf (mathematics) 1 (number) Set (mathematics) Expandierender Graph Kopplungskonstante Parameter (computer programming) Dimensional analysis Group representation Derivation (linguistics) Coefficient of determination Cuboid Circle Körper <Algebra> Series (mathematics) Determinant Descriptive statistics Physical system Theory of relativity Moment (mathematics) Principal ideal domain Thermal expansion Two-dimensional space Variable (mathematics) Category of being Arithmetic mean Cylinder (geometry) Symmetry group Symmetry (physics) Tower Chain Quantum mechanics Order (biology) Quantum Right angle Four-dimensional space Sinc function Spacetime Point (geometry) Polynomial Three-dimensional space Functional (mathematics) Observational study Transformation (genetics) Tournament (medieval) Power (physics) Number Pi Degrees of freedom (physics and chemistry) Term (mathematics) Operator (mathematics) Modulform Integer Nichtlineares Gleichungssystem Pairwise comparison Multiplication Shift operator Matching (graph theory) Surface Weight Model theory Quadratic form Vector potential Supersymmetry Network topology Object (grammar) Game theory Local ring
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Group action Multiplication sign 1 (number) Sheaf (mathematics) Mereology Dimensional analysis Group representation Mathematics Conjugacy class Many-sorted logic Square number Circle Körper <Algebra> Length Descriptive statistics Physical system 12 (number) Sigma-algebra Infinity Two-dimensional space Measurement Connected space Category of being Cylinder (geometry) Curvature Quantum mechanics Quantum Figurate number Metric system Flux Vacuum Computer programming Functional (mathematics) Momentum Observational study Link (knot theory) Sequel Product (business) Term (mathematics) Operator (mathematics) Energy level Modulform Nichtlineares Gleichungssystem Condition number Module (mathematics) Addition Multiplication Wave function Model theory Quadratic form Equivalence relation Vector potential Supersymmetry Wind tunnel Combinatory logic Spectrum (functional analysis)
Group action INTEGRAL Multiplication sign 1 (number) Water vapor Parameter (computer programming) Mereology Food energy Group representation Square number Circle Determinant Length Position operator Partition (number theory) Physical system Social class Potenz <Mathematik> Theory of relativity Sigma-algebra Generating set of a group Sampling (statistics) Infinity Bending Variable (mathematics) Measurement Category of being Arithmetic mean Order (biology) Normal (geometry) Quantum Summierbarkeit Resultant Directed graph Ocean current Point (geometry) Functional (mathematics) Civil engineering Similarity (geometry) Routing Rule of inference Product (business) Power (physics) Frequency Goodness of fit Well-formed formula Term (mathematics) Operator (mathematics) Differential operator Modulform Nichtlineares Gleichungssystem Condition number Pairwise comparison Shift operator Multiplication Wave function Weight Uniqueness quantification Physical law Permutation group Model theory Line (geometry) Unit circle Finite difference Object (grammar) Family Spectrum (functional analysis)
Complex (psychology) Presentation of a group Group action INTEGRAL Correspondence (mathematics) Multiplication sign Direction (geometry) Set (mathematics) Insertion loss Water vapor Parameter (computer programming) Mereology Explosion Group representation Sign (mathematics) Plane (geometry) Diagram Körper <Algebra> Descriptive statistics Physical system Rational number Sigma-algebra Closed set Infinity Physicalism 3 (number) Price index Perturbation theory Variable (mathematics) Complete metric space Measurement Substitute good Orbit Connected space Angle Phase transition Quantum Right angle Resultant Spacetime Embargo Point (geometry) Classical physics Polynomial Vacuum Functional (mathematics) Existence Observational study Transformation (genetics) Real number Rule of inference Product (business) Number Power (physics) Thermodynamisches System Well-formed formula Complex number Operator (mathematics) Energy level Integer Nichtlineares Gleichungssystem Renewal theory Condition number Module (mathematics) Wave function Forcing (mathematics) Prisoner's dilemma Physical law Model theory Division (mathematics) Limit (category theory) Dirac delta function Equivalence relation Vector potential Supersymmetry Field extension Network topology Calculation Combinatory logic Fiber bundle Family Spectrum (functional analysis)
Group action Presentation of a group INTEGRAL Differential (mechanical device) Multiplication sign Equaliser (mathematics) Kopplungskonstante Parameter (computer programming) Dimensional analysis Group representation Sign (mathematics) Square number Körper <Algebra> Determinant Physical system Moment (mathematics) Knot Thermal expansion Perturbation theory Connected space Category of being Cylinder (geometry) Arithmetic mean Vector space Field of sets Order (biology) Quantum mechanics Phase transition Mathematical singularity Quantum Right angle Arithmetic progression Sinc function Spacetime Point (geometry) Classical physics Vacuum Functional (mathematics) Link (knot theory) Transformation (genetics) Graph coloring Number Centralizer and normalizer Phase space Root Complex number Operator (mathematics) Reduction of order Energy level Modulform Nichtlineares Gleichungssystem Immersion (album) Multiplication Dependent and independent variables Numerical digit Matching (graph theory) Symplectic manifold Wave function Forcing (mathematics) Prisoner's dilemma Model theory Linear algebra Vector potential Supersymmetry Invariant (mathematics) Logic Calculation Homomorphismus Object (grammar) Asymptotic analysis Spectrum (functional analysis) Limit of a function Gradient descent
Hadamard, Jacques
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parameter and we know equations as critical points of doubly effective felt this is an information we gathered by doing this two-dimensional fields the mission and equal to km hr box equation is isn't vacuum identity for order parameter to facts which is like and the better equation cuticle points of interest effective supra potential and as I said this is cruel for any any equals soaring 2 dimensions for any representation the thinking is that they are changing their presentation we are changing our continent available system so when I take gage it was different a presentation To be a metal sector so different flavored rule different clocks and so on I had In a mocking sector I will get 2 different quantum endurable has them what it always will become continent rebel support think happened there in this kind of game was that if I wanted to get a head speed change turned out that something very basic something very simple some simple story with describes if I would want to get G tool a spin chains on IPO don't know but if I want to get anything that is in a D and E serious icon this script take a quick From a G 2 is also possible and but now I want to Malta this is a this comes with a sort so I don't know what I want to there is also the owner of the land and the wishes of the solution I get is what is written here but when I want to get the other solutions I have to change the problem and I it's my lectures of course this 1 gives them solution that I wrote OK so now I won't erase this and the will bring about so this is information we keep in mind that I want to move topological quantum field theory language until can explain the same thing the nationwide the issue of lot every 6 months not every corresponds to like him all results that sold as equations corresponds to the elections in his Beijing language means that you right there Hambletonian says functions of seeking like worlds there is the H expressed in terms of signal out he expanded the end so ages our adornments are functions of stigma and the formal up you get over there are the values of 101 but you have to evaluate them for solving this solution otherwise it would not be arrogant question :colon easily the most dangerous in face with hope we are not there in that language is this kind of you know there are many ways of understanding rhetoric with action and the moment statement is full you have agency farmer tries energies let's call it here because we're already so the former lover and in terms of Sigma and this will be a spectrum in the spectrum of the quantum endurable system only if signals that equation now how did it can negate language I explain if it's not a critical point this doubly effective and it has nothing to do with them now how did it get DeConcini quantum into global system another election creates a complicated long lecture was right now where identified these 2 and now we want to play games what do what can we that out of it because he I'm not going to repeat what people continent available system built and more aware not the expert of doing said but was what we learn about and how can we use the knowledge tool to expand thinks so now I want to move to the political quantum field theory language keeping his knowledge of somehow forget all this sinks now except for the fact I wrote there and a little bit digressed topological quantum field theory anthropological quantum field theory I I said that standard lower as being
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the guests was In similar 1 was what we called two-dimensional and beyond seek which is part of the story here I will say something about a foot in general this kind of let things that are topological quantum field theory that actually find many degrees of freedom so it's quantum mechanics but it does not have to be presented it's starts supersymmetric Soeren at this point as I just showed suspension with supersymmetry and misery and often buckets of knowledge and what is it how age depends on Sigma In that formula is completely irrelevant it's memorial bearing you're just taking the basis of the source of this will lose control important parties it's sum will be over solution was to get summed up now I moved to winning most of the hydrogen there is a provocative here and then the importance but on the Taurus that's justice OK so I wanted to write a general formula for can you demand surface and while I find similar sinkers much said last week that it's easier to Muses lectures in 1 week when you don't forget what you were talking about before and also when the audience is the same in of the change constantly and I think it's a different people that is completely locate soldier offer topological salaries like that as I said I defined what was a physical salary every single was exclusive defined and I also explained last time's couple of type of twists that 1 can do little but informal I register for any of them since the topological partition function is this 1 but this age by the way has this is age of equal to support is not related to energize I rights so it is somewhat were certain said 12 function age Sigma belonging to that said does apology minus 1 so this is forum surface of genius during exponential -minus some PI all I the king said that matches what also are such that be souls and indeed the said B is the solution to all our but declaration ended age of signal has actually made my will introduce to finish 2nd This is a "quotation mark exponential minus 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particle states that they spend some time discredited talking about 2 things lounge and take seat a new time 0 so 0 which corresponds to the city to the nowadays
topological metric has a property as that CAB invariably lead to BC is normalized where I right now seeing With a locker index to use metric from ISU . 4 "quotation mark so this defines canonical area following elements and I called this element page history written in terms of the contents which are the verge on Wednesday so age metric our the .period function and operator so the the sink and is also as a two-point function because we have the metric which was defined in terms of the sea .period function so this is again similar sinkers same thinkers he tied the old age don't all that's it so here we don't In the explanation of these TI or wise and just coca late forties equals 0 then there is the zip topological misery known in old formerly of the topological on Gina's G the "quotation mark Vero told me just sound or I can write trace follows this operator into apology minus 1 In this way that formalized through so that's relates handling In answer description was a particular formal I wrote here and now I have to to say what is you 0 and I'm finished I Norick described what you 0 I have to remind you that since topological quantum field theory counselors section and topological action always comes under the descent procedure from our effective twisted but sober potentials of if we calculated in a physical so affected whispers of a potential than corresponding apology collection with defined explicitly in terms of this guy I give example until the clear Plaza is to do it you fermions and there is a dealer Tony deterrent when I have to write Nelson Simon surface Sigma I have a two-dimensional Scott culture which can be coupled to some scholars functional Sigma and that's what I wrote denoted by a U 0 so in the physical so you calculate doubly effective 12 but if you have an account space there will be coupling also of this type and you have to calculate what you 0 and you 0 as we see Anderson a formula for handling operators this is generic action and its isn't too sure why any actions like that take a symmetric polynomial God all of scholar In multiple ranked multiple is here has a size and signal so I take asymmetric polynomial or actually doesn't have to report any function which is invariant under all symmetries of transformations of the field since lies in the joint representation it's joint invariant function on cast-iron on all 4 of the algebra and then the claim it is that under the transformations so supersymmetry transformational as rises on the fuel charge that we kept and we can kill any I can understand I can't take it with you was this will be the or something let's call it once I can take this 1 take the materials and will be kill something "quotation mark tool and the storm is a tool for because I had to buy due to talks right that Q has a property will that pool is a tool for of both the perfect Lagrangian I can integrate to form over to the surface and this will be my ass now these answer will depend on With the functionality of seeing lopsided and then it will satisfy all the requirements that we need to have in double-digit cold Monterville character or anything we is a question what is and I it's calculated after billion Asian From our physical salary total and you read him so we have you accelerate we integrate all all possible thinks and so on and we get some action in infrared I was there when we get to will have this fall story think is to find out and I I just said that pizza W. effective in that terror is necessary also to recall OK so I the action of the I will doesn't I will do example so in the simplest possible cases answering most questions I these changes of Sigma square but it's a simple simpler a joint invariant function on on the Great Lawn on the algebra where sigma lives so well that's the valuable told the dissents you get to the school toward local tool to see time said plus sites I think this is to form this is scholar this is to form types In general you will get this putting simplistic at that and this is the starting point of description of two-dimensional topological so is it that we can introduce called two-dimensional topological and but Michaud just to and I feel like there's something religious I'm keeping it jumping up and down so let's take that guy and let me wear quickly whole composition two-dimensional and out of the central so we have now holds the world which is traders Sigma plus side which said it so now we end observable and they said 1 0 0 always can be that's G always tool tracing the square the 1 of those TI all right I pick 1 team which and I picked all 1 ought to be chasing Musgrave's perfectly allowed observable this doesn't interact with anything so this is important for the measure but now we have defection chasing my head clients in June over 3 sigma squares it's quite that thinking things like can integrate out Sigma and you this important as multiplies water for advancing because facing musculoskeletal system for integrated calorie have to 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description had before it the same object the EU was 0 of signalize supplementing is to recall :colon because it's a it's it's only appears when you have a cards that would or could it have anything to do with it does has a it's now the office in New York New to my lectures and in it's uncensored and everything through the rest of the book it sold now mentions so this is our actions as stopped which is within them including mentions there is no that degree of freedom for the gage field so on degree of freedom if you have very much surface only cake and have it is let's take Torres again going to the polls so I right now is a double-digit collection for Torres as a function of X and the Sigma where X it is 1 of the many positive field 12 S 1 places our car and let's think ice component of because we have many 1 fields and this is exciting so the action actually the functionality of from 1 of the meals a gage field scholar and affirming on 0 months because arrest of overseeing disappears independent of points and everything so what is the type of the typos attention it's nothing more but to quantum mechanics was P I X built With the PIE His contributed to the SEC side and is it will be doubly affected will be the Sigma play exists multiplied by the time were to observe what you get this is equaled this is where it all rolled into the lower signal this case Torres all the WBC knowledge times FIA which now I write his dean of and the only time do you would you like keep here there is no space dependence I take alluding to grow all arrests 1 so I get that disease section which is always there is equal PX .period it is clear it's two-dimensional general degrees of freedom on a monogamous care I wrote his appraisal and this this is a time direction here is also material was a time and since it's a Taurus it's also periodic then there is another term which multiplies variance and this is important the fermions come with this term so for Munich passing the real will give the determinant which is evidence that the tool W. affected this admired his signature this term will give exponential-of of minus told you 0 in a Shh and the wonder wonders obvious because I'm integrating matches since I've got analyzer mattresses get wonderment tournament I'm explaining due of wonderment is clear determinant of the derivatives of doubly effective in positive power comes because it's a very Munich and develop science and it's a positive power then there is an explanation of mine is 2 times you 0 signal of which comes from Ottawa times you 0 their region and in the last term and the all-important think now use that see the stern P X dog is defined X is defined shaved by 1 because I'm on a circle and me when I go I inside lodge you want this transformation XI will be shifted by up by 1 Egypt's I will be shifted by 1 so excise luminous and if excise when circles and PIC vintages because I have exit quantum mechanics when excellence on the circle than going to moment should be integer and equals the integer Is this a question at the moment I calculate z topology collection which is written on the blackboard how I cope with is a different story I know that's the answer will be some always solutions of PI equals integers but now closing the is complicated in two-dimensional young wheels which was somewhere here on the blackboard work over there PID policy to join man that signifies according to John and seeing Michael seated during two-dimensional young males was interpreted that we have sound all our highest wage representations of read-only Grove which had labeled by set of integers so the set of integers in two-dimensional young males interpreted as a representation of residents of were integral and all this kind of thing these were highest weights of the individual Unitarian representation now these relations saying Michael sand gets replaced by complicated won the WBC Michael said so meaning of the double the singer Michael Stipe burning Porter is the same as their highest weight present so we have the formation somehow all all the salary of life with presentation In moreover it's not only that but also soon but we will get to it so sold the better equation that I will now do Verheugen example quickly and we will see it sea and topological serial Loveless integral model is pitching that's actually directly official for 1 2nd I want to ZC like you I don't have to be and now I'm coming to Turkish black you don't have to be finally dimension I think this is a case when I have the remarks of operator which the other the 2nd solution is constructed from the 1st solution from the rusty and I don't want to go there but that was the moment I want to 1st rides upon a mail 1 and then I moved to the case when it's not funny and then I claim that the 1st important they planned to visit is a box that equation sold with each spatial like and for each solution there will be interpretation and instead of going to the negative spins the negative power ourselves 1 polynomial I will what my plan was to present exists his little bit organized on the way that I 1st study every single which is what I call complex situation in the noncompliant so I said that term XXX being chain was standing close to a common tool two-dimensional so averse to mitigate celery with some final
dimensional representation of which was reducible but some some representation the 1st think the need to I need to explain is that I can be asked What if I want to get yes what I can do is its expansion Jane I can change the gage group and I said What sorry I can't change the spin growth but they're right here single gene I can change finger or I can change some parameters but eventually XXX means the restriction already What if I want to do XX well on series we have to take this representation to remove international and specifically we have to take and coastal city dimensional superior Minnesota on a cylinder times as 1 reason radio and now the at an equal to three-dimensional supersymmetric gage story on this thing is the same as this 1 two-dimensional Swiss With the representation is some of our ends where n is a consequence so what is this thing takes its international celery expend every field in three-dimensional so it's cold to fine so this was instead dimensions any field expand legs said you have now infinitely many colds declined than 5 feet 5 and in terms of the 5 end this three-dimensional supersymmetry Gates Oregon actually two-dimensional suppressor mitigate story it's easy to count what does a number of Sopore charges it will be the same number of overcharges but the representation on will be same as before but each failed to get the label this action and we have to watch what is this label .period this statement is that this labeled as the Tichfield fine I will have twisted last proportional to that things which is a colorless a climax and that client lasted must it's clear actually is at no global growth global symmetry girl from two-dimensional point of view contains sheets In exiting right way distilleries two-dimensional and in-situ sewing circle we have a symmetry of shifting In ex-CIA and from two-dimensional point of view this is a global symmetry group this is a group which acts on the face of fields right when you shift Ex your off-field and for each global group that now this is off on right because this is a the shift so now your masters are associated to that'd be everything is a previous story so we completely comeback come back to the previous story except to we have infinitely many representations and the claim is that the on his or her appeal it is proportional to are actually when anisotropy goes to 1 are opposed to 0 so when our goes to 0 you go back to two-dimensional sorry there isn't emission-reduction than locals declined molds the two-dimensional story already studied and were proved that it was XXX but I said that two-dimensional celery was took ,comma 2 supersymmetry is giving minutes said XXX being chain set of accepts that changing depending what matters sector I take their this 1 will give me back to these when I extend reduced to 0 which means that see XX said has to go to XXX that explains and this has been proven to and this is true and W. effective has been calculated and forex sixes and coincides With the young function for success no interviews will be the function I will talk about this little xyz model now it's obvious right you have to consider 4 dimensional salary on assumed times to talk to a Taurus has a 1 Motorola while the light tower which is a complex 1 and it's expressing terms unusual trapeze XXX XYZ XYZ has the real parameters which are the owners of properties and these 2 real parameters are parametrized ways away a Dr. John description of paper 1979 in terms of talks 2 and that's exactly same activities to tourists will have in 4 dimensions and four-dimensional so the times the is a gain are which is infinite dimensional now double sound of understood Torres and we included we allowed such infinite dimensional representations were worth studying two-dimensional that's a game W. effective can be calculated which will be now again double sound always is double codes declined wallets and it coincides with the Andean function of X X X Y model that's has been proven now what is the Andean functions young and function is it's it for every this were you for every quantum interoperable systems every quantum integral system we know excess what is called an equal comparability the better equation please always critical equation for some function well into the millions of people called the and folks better equation is an equation on some number of variables it's not as many equations as number of variables so we can override always the better equation continent available system as an equation of zeros of 1 4 the Bozeman equations where the eyes of some renowned as equals where numbers is a set of renowned dies but just as many nonetheless as equations we construct our toll-free 1 form in our have learned dimensional space is called the longer and in every example accepts this any equal for interoperability that some of you have been working on the state and
observation that DVD equal or locally and this is all locality of course we're talking about being tired of flounder the of why the land I and this has been all observed in every known example except the game the solution to an equal force moves into global systems of people constructed OK as they just explained here what has been proven case-by-case Was that number effective all 4 of those stupid and minced celery coincides with the unhcr function which just blunt identification signal equals slander doubly effective Sigma equals 1 of them has he is included in I all Europe I don't include it in Winehouse household atmosphere so I don't include in Weiser migrations BIG cost 25 per cent minus 1 and that so I have to include this function so 1 moreover yes and moreover the age 100 ends of so all eyes of stigma of Seagate's salaries is undesirable side said where identified with the child the energy as far as a quantum 120 times that people calculator this is defined in a .period introduced systems just the convenience thinks it's a people Palmer tries eigenvalues in terms of some of parameters land on which they find in many different ways and then there is the statement said the market expectations when the Overseas operator Art of coincided with energy but this has been verified case-by-case 30 complete salaries that we could say when dust and everything and that's what we're looking for so now I want to consider so I listed these things because I can't spend time concentrating on but this 1 I hope I explained now want to consider the simplest possible case when we have both topological quantum field theory description and some interesting troubling their operators that I think are worth discussing but let's go back to 2 dimensions for gets this kind of xyz and so on and consider them my old example which was called Maine example except I will take a while to be equal to 0 which means I consider just representation to the 1 I joined all you and I call that 1 of joints to verify now I have an equal to Sorrento dimensions and I said so pure and equal to which means just did not let your equal to plus I had won Johnny and I made the comment made last this is a name that we will be using later this kind of service called and equal to starts so I'm equal to star salaries on an equal to worries plus dropped 1 I joined massive multiple it's the same name it's called in 4 dimensions it's called and people for now there is another properties at this kind of sewers had the salaries resolved so any call for sours so they have twice as many charges I mean that in equal 4th soaring cancels a field ,comma so they have same field content as an equal force this was the the whole thing was that of decision to said the of this authority xyz then it'll 1 pieces such sale or is it when you go down to Torres to get an equal to and the slighted by the way as strange name it's an equal to incident this year they report would "quotation mark was not always so far there is no way to get in what 1 of the categories Alexis II goal dementieva reduced 1st and dimension reduction I identified what type of story I can have and then I say that you find now have Toro in local to climb ordered their one-to-one correspondence to xyz it is an operator with some signal is still some capable system but I don't know what the name of the man he established by what's what sort of information that you just can't see how it would probably you I didn't know it was just the the work on the structure of the state of of this is another notions that advice takes in 4 dimensions of the is defective supra potential in 2 dimensions to visit that's how we will have salary for the four-dimensional salary any close ones it will still have the people they have to take to to make to a clear that let's take it all the source was when I'm talking about for emissions and equals tools broken by hand to 20 quotes 1 so this Saturday X was the use the I don't have to a placing muscular those here earlier this year .period for .period action that someone would want maybe was let's talk about that you know I'm getting a bit tired so let me up to some points on make it's quite possible actually would mature their coveted in surface if I can interpreted is I I need all the guys to be masters I knew the recent remarks in the theory he had the expert nothing for also expanded by the way they there is an important thinks it's the people who do xyz model citizens better and that's a tough to justify their world for xyz model it is an equation of that type that was written here and there it has or satisfy certain model of properties so there is this statement about xyz being changed to better equations are somewhat lower the function of the ratio of 2 the functions and it does not clearly satisfy what larger property on list the better routes I think some up to Xerox so there is a condition that better routes with some of the hero and then they left inside is Marglerite inside model or instead of instead of products instead of product of seeking my mind is the 1978 less and or something we have to do functions and they characteristic and something and the when when you write equations I was writing the recent problems with light which is related to the facts in xyz model With the
ground state that actually needs to be constructed but that this summer a combination with some teacher terms of Syria the usual grants supply constructed by Bibi Bibi have Tinkerbell operators and that would like to properties that I just said about the forward thing this better equation that if I remember correctly if you write Kansas a language of seabed tunnels some of Sigma I have to give their clients of conditions it's very easy to read translate this store xyz model on these times Toros that being good any "quotation mark being an equal 1 in 4 dimensions of that kind Torres if you start by putting fluxes on 2 so you can put for example to fall flotsam 2 and so on general salary and there and the conditions of modularity as as far as I remember is equivalent to to conditions of the anomaly freed freedom in this for the four-dimensional gates only when you have the ground ground fields influxes on too the field like a put to be filled to fulfill any program and they have links in this lies among its entire new subject lecture me about in this this that's why his evening in equal 1 4 which goes Lexus you observe something on their cylinder time stores and you're happy and now you start playing with 4 four-dimensional solely on his 200 thousand dollars and you want to explain everything in themselves xyz model and lots of interesting things happen and that's I'm not studied by anybody I don't know anybody study OK so now let's make at that because they're not aware of any call for soaring and we're going have actually many many things in general we can take their which can equal toll but are really an equal 4 and then break and equal 4 don't by something down to an equal stole and in this particular case this is done by Eddie Ingram super potential but I will call there super potential "quotation mark die of Sigma up which is one-half trace Sigma square in no no reminder this was the answer 40 boys somewhere here just years I said the two-dimensional young wheels was like two-dimensional me was its Shiite edited by hand tracing square and that was widgets explanations it's a two-dimensional Lorena two-dimensional and will actually has to have these completed naturally and males and that's what I would do here also so I would take no he said the weekend taking in equal to or celery and adding tracing was quick I would take any calls for celery gives a mosques and one-half tracing must square and I will have the group you and so my question is what is this sobering obviously it's not XXX the change because I don't have I have 0 lengths completely I don't have a stranger but there is something similar to what that something is and that's something Of course we determined text by calculating it's better questions it's vacuum equations and last ones will collect immigration as well as a question so what where we have seen this that there are lots of other questions people can ask and I can try to answer like yes Yukon Flats pinching but did you construct sustain itself or used make a statement that the spectrum of pinching Hambletonian is the same as a spectrum of rocky that so but the statement I made now do you let him come and tell me what I want to see this thing where is the speed there or the can in the construction in two-dimensional gage showed it to actually see this thing and that's doable but I'm describing here it's very intricate than it's actually possible structural for OK so they'd better equation now knowing that model which is written there which is any call for last mask for joint plus 3 level but then potential the question will be exponential-of I Sigma is "quotation mark products from equals 1 too the mind-blowing forced and do what you miners signature minus and here I would stall the
allotted so I would say you know what it is this is called the article sectors so now I detected I rolled better equation like immigration now I know 1 for a week which indivisible systems and particle sector all the nonlinear shedding their equation this is going to feel so actually in 2 dimensions in 1 plus 1 dimension but idealistic teachers in particle sector so it becomes mechanical and that particle sector is described by the police and it also in that and particle sector it has a quantum-mechanical description in terms of something looking like that take any particles X 1 2 X and moving on its pro have any particles on circle we as a 100 Tonya To be minus 1 have from 1 to 10 weirdly exercise plus the climb some final people generally don't function of excitement 6 Jerry and that is m because I see the could so this is an identification This isn't known better equation for particle sector of northern yesterday equation which is equivalent to end particles on a circle was a show bingo operator being legs so if sequels they're all this unjust and free particles on the stock and when sequels there are there should be two-dimensional and rich by the way constructive so too much of a meal is he but it's not the sort to the formation of the national and the measure actually correspond to single infant but when I come back I discovered so now we're dealing with a situation where the infinitely many rock it's a fine the measures it is a two-dimensional supersymmetric gage celery with financial in many fields but has infinitely many supersymmetric vacuum because everybody knows that this system has infinite incentive better equation as infinite so when I come back I will start to resist and connected to the kitchen system and that will be my way to move 2 Hitchens system from the 2nd half of this lecture and next next week I'm very sorry today was a little bit got kind of jumping around because you know when you finish in 1 place and you come back and you realize it's is of different people and they forgot what we were talking about last week is a disaster I mean how can I warm-up to get this former I don't you OK in 1510 minutes maybe we come back 10 15 minutes 1st inkling all that we can asks a question former which would especially we integrated In this case and diagnosed in mind that there was a generally I rolled seeing BPS equations which looked like which takes this connection they could construct a curvature and there was something here please dependent upon complex conjugate to sober potential and so on but this all this is 0 except we have this the 1 fielding a joint representation which I qualify and what was American side in the republic which equation will be just commentator also flies in the face of wire and then there was an equation also what was below Zebari off they need Cottle multiple ads that I had in metals sector and now I have on his knees to the sector and that will be sold the sudden appeared equation integrating lowering engage celery in that set up general equation April relational live what I had last lecture and although people are not here maybe someone watches songs over the Internet can be useful so general equations was for every matter multiplier which I've been altered by X there was oblivious equation that topologically twisted so was integral or equivalent is a vacuum sectors the physical celery defined and this was the Zebari because this is not like of globalism bar of a acting on XII In the Matter sector pulled minus G. I .period borrowing the Dudley stock of exit which after 2 is become 1 4 the excised and then there was another question have for is bar or a pleasant viewing of electric supply was 0 conceptually the system In a divided by section today's growth and here and the gene was defined to me was the algebra renewed momentum for sympathetic form coming from Kalimantan and GI Barboza was a killer metric it's a lot these equations that's the place by In our particular case a for salary was lost but the question is the statement is that now we have integral always a model estates of solutions of these described by quantum endurable system of and particle sector of nonlinear shifting the creation what solution what we know about this more than you should figure creation in part because the 1st think when all about this Is that we can't exactly solved we can't exactly sold the wave function equation age size equals size and this was done I think in 1963 return the shortened on useful to people but
by no means end meager and there is the Paul Hill and think about this being a Russian I always have to "quotation mark passion if it would follow the American Paper sold Sicily and this is a very famous the example and Brazil 4 and think so that actually there are higher Hambletonian I just wrote in his 2nd half of and there are 101 ends are important in terms of something called Dunkel operator solved that it's not only salaries and use nice formulas for and February similar in parallel formulas in the shutting operator fill for the spectacle of Chilingarov the because if we will need for future to generalize Orioles and these are following sold taken as a differential operator for plus tie em all with some equals from plus 1 2 that function as excited when the situation plus 1 times fermentation inside so we have here SIG which is from the wild rule permeates positions excited here and we have a deal by which is a shift in the form with 1 parameter and it so that it is not there this is just the SE is meant to get as a result she at this would be called after while this was probably no 1 quite correct OK so this is a defamation now of offline while growth and has a name so we have to generate to type generate resists and SIG and this is called doubled "quotation mark Hey algebra at the generation so there is obviously something which is not to generate double opined had killed and this is a degenerate and it has only 1 parameter but so claiming is all coming onions all of this in durable system can be written as 1 so this is case Hambletonian one-half sound this is using the issue by the time we got to the the book is all about I don't have to the generous story what I have to to do is just to be on the side of the law of by acts so I have to that's your thing the multiplication by X X shift and the that's what you're saying probably within this discussion I adjusted work with clients but I am saying is it's name all of these things is that the what I just construct he is in the center that's Elaine and probably year uh well 1 1 permitted the formation of a final Welker while 1 parameters I was told by you wanted introduces his die-hard degeneration so he claims this is a degenerate .period will grow reckoning formulation but since he explained this to me when I get to see more more more serious will discuss with that's what he calls the Gerrard . introduced . so you knows what it is but important things that these guys are in the center and our age is so this age tool it is a square please 0 1 have 12 sound 12 square and so that's what we need these will have generalizations when we got the tools of more complicated example and then there is a formal around for good explicit away function which is common Eigen function for all of them now watch I don't take yet on a circle UCI wrote to step function so these 1 was circle which means it was a periodic 1 now I take it on infinite line X is on Our ones and these people I just said their Lehman current text explicitly function explicit way function of this sink is useful for us In the future discussions at least in a way get an aggressive and I explained more complicated systems and irascible and I explained this particular example so we take elemental of permutation group were some overreach with product of finalists and Jay signal called W -minus signal of the route plus and that functions Texas minus J which means that the 2 would just order them I don't think this is just 1 divided by signal the Brunei -minus admirer of times so this is a perfunctory exponential of talk by our sample were said Mark W. that's thank tho claim is that this is exactly where function forces been value problem where the energy His asylum Sigma I was politicking so for example for the age just Sigma square some over Sigma squares would be energies and this isn't exactly function of the nowhere is the better question before I go to where the question I want to live here in this part of the blackboard the normalization so these are not normalized way functions under usual measure on R 1 which is now are 1 to the power and we haven't particles so the normal normalization is so we normalized Sigma 0 signal prime 0 Bloomberg integral it's 1 of the or and all save are sitting on a 0 sum signal prior to the World and this is not equal to 1 this is equal to some signal jeered at him a quality product of value equals 1 2 and the function of Sigma mindless civil pride so after this fructose and Janie this normalized at a function normalized wave functions as it says they're supposed to be and you've seen actually it's something that will become please familiar overdue for the word by so this normalization eventually will be connected tool To handling operator still have a determinant of food to Witchazel doubly effective
and that will be connected to the trundling operator will have a meaning normalization is there and bends and better equation which is written here I explained in a 2nd what to do so it is clear that this normalization show-stopping and now once we move on from here to circle from the finish line to Sarah we have to request that side satisfied that it is is the condition of women will buy if we shoot X by 1 so when we moved to the unit circle so let me right here PCI Sigma all facts to be normalized 1 which is 1 worst square until you've seen 1 thousand slicing loves and when put this condition non-exclusive formal I have written here it's obvious that in Canada arising from here you shift expire 1 you create from here in the near term and you have so the longer all of these when you pull up in financial so this is by the way just for simplest just little with this piece of asthmatics which is written here our model and we have the logo for asthmatics as as a phase shift so general statement is replaced function excitement accentuated by periodic function which is somewhere and sidelined security plots requests burial disappeared from the wave function and see what happens wisdom for the waiver and uncertainties where function with periodic that equations so what we have now is that we have quantum many system a single 1 Qantas many water system where is pairwise there is a function on end of the spectrum it saying there on open system class restrictions on the repeated use of the better course this is the knowledge that the turns out to be extremely important I can't it possessed the questions in the English because of getting Cardinal I will make the story much harder or take care was to get to the statement that Beck equations in that sense I mean restriction on effective and so on is now connected to the producer and that is that city will become Howard when we got to more complicated examples but whether equations that has appeared city here and that Warren question is that these acts which escape is not the same man which was here the weight of we had .period visited there was a period city in this problem also that's how I got movement Britain which I requested but it is forcing 100 and the goods as composition .period "quotation mark so I get conversation there From the same required because what's written here this equation is nothing more but the exponential for I at times the W the is I equals 1 if I take a lot of these and moved to its financial here and that W is the youngest function for this intolerable system which has been well known for a long time so very important open question is what is the relations between these variables x and those variables because in a sense we know what is the relation between the variables W D Sigma but which which we have fewer and variables which are a signal of this essentially so she must use the same but exercise not to and this is open question we just have a coincidence or subsequent to that looks like it's more now I I want to make an important commentary that's the history representation celery what recalled degenerate .period or maybe some Dr or something clicks it is connected to the other thinks that may be in later we have to use and there is a statement I'm collecting certain statements to to use for other problems and that statement is about functions and solutions to certain quantum many water systems connected to the representations so this is the most simple quantum and would assist them in case I understand that is connected to representations of some of the break object but this is not unique many Azeris and knowledge to describe some of them before I moved Tuesday for dimensional gage on annulment background so fresh statement said that a function roads has something to do with them what's called Holly toward 1 another how it would polynomial itself as was explained by the 1 Dutch was magician 1 year in jail story were forbidden from station into sewers and 6 the Holland to itself is a solution 2 different solution of this problem solved this problem can be instead of 2nd order differential operator meteorites difference operator of 2nd order which made becomes this 1 so there is sometimes called the I'm the young system which is probably not correct because it was sold by these people and younger and younger 1969 young and his brother In the most famous In the discipline probably the worlds of equation in this philosophy and this same paper words they introduced thermodynamic bedtimes that so all these young young system has a discovered generalizations and how little point itself as was shown by by the gentleman is connected to that 1 but the ones that we are using here is limited and let me describe what limited to go this holiday .period Nomura's lasting more but G Ellen human beings circle function and there is some this informal flat which I probably will not frighten us but depends on a bunch of parameter called me and under his kidneys and the peace an explicit form there is some single I take from you want to committee partition of of length that was then but but but but but but but
and let's try to play once we have and one-time and tools we have to and then the function itself this but want go so this is some product over I have 1 minus the court declared 1 industry and our our son one-liners when 1 to support our banks of the presentation ,comma Ga promoting exists the other variables Fernando 1 time some product so now imported statement that eventually I wanted sometimes singled to generalize it looks like it's working here we go to continue to limit let's call each other Ltd .period to 1 and it seems products Arkansas the Wikipedians a product and bodies were not on products what so I have parameter peeing and I have news which are petitions have so Almeida and I have news which are like our coordinates eventually and the peace product here from 5 1 to .period I believe primary source and on there is a some only in symmetrical there explicit for now that's the only important to formulate an explicit form for that it has to this holiday with polynomial and that limit his legs take me alive and right some continues will will excite wide Wetzel London needs to be exponential tool provides excellent time signifies that would needs and sent excellent was 0 where TO where the patriotism that's what officials told pipeline and Epsilon and send absolute to 0 the claim is that when epsilon goes to 0 this becomes that means many Caraway function it's explicitly becomes the organ function I described here that's just when you plug it this leads to the status None of can number which reflects and his number of soup articles and particle sector so what I wrote so there is claim his legs this equation has its discretized much which has been studied but also into groups then discredit that ration has explicitly holiday food polynomials which are the wave function Gallatin was mills from my Rome side Our jail and QP is almost vertical functions so this is the same as what you do when you do this political functions when lose orbit met divide by Jill and that you and I are divided with this that which I'm not expert but 6 explicitly take a former law and that formula you make substitutions such that wave function will become wave function of this operator funny way that substitution also takes P 2 rose to 1 so please exponential effects the basically what you do don't like we usually do it takes function analytically continue everywhere it's possible takes a substitution intake limited to to the will of planes this limits later that's why I'm I'm discussing it here and this becomes In the solution to that came this schools some kind of forum picture which motivated to he aggressive inmate told next Singh said we buy will be talking from now on and this is following the board needs to be fixed so let's take what we know about half an hour or so let's take situation was free fermions so this would be a two-dimensional males without any heating systems that I've been described here so when there is no fight when I answered some 2 was question here when we turn on 1 parameter which is this parameters will introduced takes failed before we turn on the Bermuda and we somehow go to northern shedding their system one-sided there is another way of deforming at 1 end users knowledge becomes point for me which gold-colored Jerome was there Sutherland the formation this is now this was connected to the delta function potential or let's say 1 parameter and this can to the formation with 1 0 loss seems square footage to gain amid all I'm discussing now on infinite produces conditions or something so with deformities like and then there is another 1 which is I can this rule you sir Maoist defamation and the 1 more 1 the this 1 has 2 parameters lets me :colon Q indicating this corresponds to his fellow in system is relativistic version of the College of was Southern so this is not optimistic many quantum anybody's system there is it's that Asian dislodging postmortem what's important here is existence of the and water trees and ignoring Ashaninka 1 which is sending kids this young young system or enabling the system is a Q equals 0 forces to permit friend call Jerome or Sutherland is Q equals power renewal toward a new user could efficiently in front of potential and then you have to send Q 2 1 simultaneously you have to send 2 to 1 so it will have this 1 parameter and that's what we colleges Sutherland Mauro and that there is a picture and work in this feature we detected 1st doing them the calculation for Hicks bundle for earlier kitchen modular space he told this equation do I like it turned out to be that :colon so this particular 1 that's all I needed from the picture was this as this 1 parameter 1 and there is another piece of self if I go back to this equation I want to describe 2 interesting limits of this question In 1 interesting 1 which is to the point and this somehow his family the awkward is not an goes to 0 really but as and goes to infinity and so on and goes to infinity this is actually it was a classical limit for this 1 it's not goes to 0 which is cause a classical limit but when and goes to infinity when England goes to infinity this becomes minus 1 was the polygraph and you have a statement that it's for national of exceed is equal 1 is power and and this is the preferment .period the when and goes to 0 in now I saying why it's such a deal so true to win any goes to infinity these fly this field fire which is fixed field for
which end was a mask is infinitely and in quantum field soaring where this is a matter field did the couple so the story becomes without must goes to general images embedded it because it gave becomes must and 1 1 last again cannot integrated out so free variant .period III and goes to infinity limit clarified the couples and we get into the measure the real story of Edward when angles to their image what we get is that the the agency's them excellently ever mindful that the solution to this equation the space of solutions can be equivalent to describe as pair fled connections for connection on a plus sign 5 equals 0 divided by complex if occasionally engaged the little spaces of the same in 1 2nd for prisoners explained why this was spaces of saying takes his equation this equation is Nov complex with connection its involvement on the complex featuring rigid group so gage group here has a directions and the complex direction so you have to gage fixed to study these kind of equation and you can always get fixed by imposing the condition so so here what we had this equation after by plus is the same as distinguished as plot as an imaginary part is this equation glass complex going to the beleaguered a part of that equation you with this statement in reporting the news the company is going to get it to gage extent can introduce gage fixing to bring not blood 5 -minus Lobel bar and Jericho so if you wish gage fixing potholes a gage transformation makes this equaled that this is simple physics for pollination why 2 systems of the same and must equal to 0 when we take muscles this fighting "quotation mark was there if there's nothing more but integration with this model space where Pfizer muscle but this is the ill-defined because it was last phase of the complex web connection it is not back so intervals will diverge and the point where we introduced mosque for Friday was that we would make the bed direction in the complex their direction to regularize by putting Zagallo see Annex financial minors must firefight power and that would convert so angles to the limit is back so had with his calculation for good whom fled connection the real connection was that it was somewhat inadmissible representations of wage growth so we can I think it was a 1988 that is the sound all over Europe of gage group G now there's becomes questionable because Jesus replaced by GC the complex complex indication of Citigroup and what is sun-dappled sewers filled and someday but I wrote was guessing what will we explained my calculations was an aggressive and more from 1998 in this language so and goes to the limit is bad but there is some explosives now what about people who studied that equation what do they say well they says it then goes to infinity limit equivalent to page embargoes was and so it is it is a good 1 sorry for the no at the end goes to 0 image that's what they're saying this is what for this man of Tottenham and goes to 0 you have to send entering such that some combination is finally going for broke will boldly and and is fine and I tried to remember and then this is a cynical so sent classic is not set this and then these women goes to infinity limit and what I describe actually gets very complicated is connected to the infringe instruments but when an goes to infinity limit this has an extremely well studied solution that it take when it goes to infinity for solution will be Sigma jays are one-half times integers and then you perturb around it and there is a very nice perturbation souring in 1 or more on in the storm has been proven that for every set of integers ordered set of integers there is exactly 1 solution for signal once in my head so this is what they call completeness since this set of integers ordered set of integers literally denies such said and warned is more plentiful and describes isolated the civil representation of a unitary grope what is claimed here that for every highs with representation of renewed our growth there is a solution 4 this a question so this is a case when it's well-funded and of hazardous various limits now what about going going Tuesday other picture which was a picture of a function so I rolled there that there was a how little polynomial that was available goes to 0 in how that feature in this picture can access that I wrote on his blog I think the grand entryway again people who studies is kind of McDonald's polynomials know that so long as top you have something to which Scott McDonald polynomial depends on 2 environment rescue and knowledge was a blackboard and the clear when Cuomo still clearly made this becomes volatile and equals 1 people former here that's exactly what we had few close to justify our new fuels to want to go to want unchecked polynomial here when goes during this becomes me on sure for functions which shows a close function for representation of an From we think signed this is as a rule saying the 2nd in my talks that I don't much distinguished but to do this is 1 is amended so these pictures now one-to-one sold the London will will be relevant for this and so on but what is missing from this picture it is when we instead of this 1 or this 1 we consider it a digression so this was a situation when let's say on levels of ways that we were working In the 1st the local saying that it was released we will we will do better than that we will produce a good question is to send 1st if you ask me to construct the wave function right 2nd if you ask me to construct a spectrum of the quantum Intercable system now usually would say is that all you can ask a satisfactory result function whole lot like we did here I presented the enabling the calculation of wave function then I take the pair discipline could to discuss spectrum so what you would expect is that's the way I would now proceed for the McDonald's and everything I will produce the solution which would lead to the better equation which would cover the entire black instead of that what I will do 1st I will eventually get to the but 1st revival looking on this example I will try to find a quantum field salary for each corner all diagram set comes from In my way of describing the quantum global systems I will construct supersymmetric Young Mill celery which will have the vacuum described by by all means buy some quantum integral system that I know what it and that quantum durable system we will have the description and have so the attitudes of potential so by that equation or about that
question when I had to say it's at 1st was I said that like you're given by effective use of super potential and 2nd they said that there is a vacuum identity including battery-operated fireballs languages so in 1 of those languages I will detect which quantum Intercable system I get and then the other will give me a spectrum now it will be produced without giving the way Fox 1st and then people would complain and how can you make such strong statement without actually presenting the wave function but I will say that the nest sorry if I've properly formulate the question since textual is independent in which coordinates you ride the wave function on a statement will be that I in this kind of Hubert space and here is suspect without giving way but unfortunately because people ask questions like cities there sometimes very unpleasant and I don't think questions I mean people we had to produce a wave function so we would approximated the wave function somehow and argues that there is a way function which can be calculated the isn't this way and will do that Spector and we will do that for entire that yet we will lose but at the moment I say I want to consider the alleged aggression overseas were Sutherland will be eliminated so if taekwondo will contain Sutherland doesn't it it's the ability to grows announced 1 which will contain was announced Everybody knows that was announced has something to do with McDonald in all these links the blast but I will produce I will produce they the question of thing with exact spectrum and I will verify that this is exact spectrum in some correction but suspects that correctly that space will be defined in terms of differential quick this is this logic is extremely important this logic is that understood that they recent crashes are fortunately I'm very light had was my teachers who would tell me as a domino we're its president over there said that in all my right by wrote books when I was young bought this exact sold or what is the problem that your soul and after long discussions discussion eventually understood what was the problem that I'm sold so it needs look what happened here I was I don't constructing Seaway function of with saying that integration always immortalized pays off he agency system set up in such a way as given by an equal force barium will soaring to a dimensions we said super potential to the best of potential 1 one-half Sigma Square New World producers the exact spectrum all this quantum mechanical systems while the bees 1 onto mechanical and call ,comma Nikon functions and so on I predict that I did not complicated but then happened that it actually gave it correct I am informed that the question that these coordinates in which the wave function is written here In the young young system angrily bringing the system is unnatural from the point of view of the gates so worried because Gates Saori likes to ride the wave functions in themselves the uh moron drummers subjugation a Lockheed indicates short initial case there when we have few friends this turns out to be saying cordon of this exercise here and that exercise in the case of the 1 of the characters of the representation of central group of the same but once you leave once you depart once you 10 that because that are major masts which was a twisted knots which I said also the current parameter for a global growth and I turned it down they depart and this is an open question is not understood how in this case we know everything and we verify conjecture that the gage calculations of the Eigen vectors and I get from the values of Zacata loading operators actually do produce correct spectrum in a correction but space all corresponding go into cannibals OK so in my view will be Ingle person working in a gage service I would say you have to be insane not to believe that this will work for every case as long as identify what is my Quantum Intercable says that if 2 billion retrograde Russian they say islands of the person who doesn't believe in progress that this is not going to give us the right answer but to get through they said an aggressive and I had to verifies that we actually have to solve any particular juror mother equations order by ordering verify that our statement is true or not but that of course was sold at up to very few of us like 3 for does give the I can't get so let me know more 2 of the most interesting puzzle problem my lectures is that C bonds were determined condominium would assist them the CII described his corner here 2 I'm sorry I didn't discovery well this holiday for Commonwealth which is not my but that the conflict matching of this but I moved here and jumped immediately to lift tickets because the case ,comma Severus England he comes forces from 4 dimensional I agree with and so on the models which are written on the blackboard in the right hand side he said those models finance sigh situation correspond to some of them are our friends discovered correspond to the library stories so have to move level From now on two-dimensional models dawdle competence occasions from Seoul will disappear and remove tool for the National superior males and algebraic interoperable systems so in the mouth 1995 dilemma Gorski creatures the euro notes and Marcia Clark and Martinique and warned and then the and wages gave very beautiful explanations of connection of sided with changed solution of supranational soaring for any calls for superior males for the end of algebraic classical right into the wall systems that I need to review have 10 minutes left so I will just introduced a basic notions and will be done and when there is so vibrant and written in 1994 founder a beautiful on dissolution tools over and well soaring for election was 8 Supachai so sorry was a supercharger in for dimension is end of the closest cousin of celery was discussed in 2 dimensions so we have been told to go forward superior and mail surrender it has a property so in perturbations so it is 1 of the exact so anything goes in coupling constant effort to the views it is a coupling constant but was there is a perturbation salaries the 8 charges in 4 dimensions but which still rely on the exact sold then you have 1 little perturbations salary plus the instant time blessings that ponds and even then see is that the because of this the higher number of supersymmetry there are no on Tuesday remember when he is sentenced him to life it was clear that situation is very complicated you have instant isn't always interact with each other that is a complicated mess and it was fun slots but the following charts for
calculated in the late 70's someone search using announcing stint on some exam up in signal models in 2 dimensions and recovered to answers but that what which shows that in order to get for advances you have to To all this complicated interactions between the now being an equal to Surry there users an instant also instant don't want them not to and there because of home offices longer argument around but it's not important for us what is important that observation of the guys low-energy effective action the exactly alike into days W effective produced low-energy effective action in 4 days it's another thing which is called a prep potential now in in 2 dimensions we had 1 invariant function which recalled doubly effective and we had to construct 2 form out of so that affect it wasn't finance functional is the linear algebra all of the gage girl and we had this procedure all of descent procedures that would take his inline function ECB ideally would leave its key or something and so on and produced 2 4 so in 4 dimensions for this kind of salaries claim is you need to know 1 homomorphic function on crack and always in the algebra holds a gage 1 invariant under the a presentation will function which is homomorphic because the funeral also but I this color interactive multiple complex field 1 function which is neuromorphic and you have to construct 4 and this is effective Lagrangian so effective actions His athletic into dimensions which was a tool for understanding from the effective interest in Super potential descends wasn't from 1 object which called the prep potential said Woodward and potential and there is a descent procedure so deal affair is or something in Europe it 4 times and get a feel for it on the need to know this for me it's so trivial just 6 points as his import market sometimes claiming is and exactly like here in two-dimensional and equal to summary we had to calculate from you will be sorry we had to call :colon doubly effective I hear from you listen and you have to calculate the back office that's a compliment but whatever you do there will be I was so you take the will here whatever so it would take several Dublin With these guys observant I finish on said this had available to them has meaning In classical algebraic Integral Systems classical for collection and it's also calls their privatization so we have a statement there was a point of ill-suited for the miscalculation led to someone home or metamorphic function f of Phase I bring with described it and so on and these people mentioned notice that if the potential hazard regimented meaning in a classical Hamilton insists complex algebraic undertone system and when I come I will start with this and I will this shows that the 1 knows examples these people had also complex algebraic endurable system is in the middle of this and we have to question of what I wrote here is particularly which means that if I know that f afraid then I can identify with elliptic version of what where his roots in here now and then can ask questions very similar to what I was asking before but instead of considering disease so respondents cylinder times Toros where I said that they would to it's wise immersion and when was it said that influencing will get something interesting so instead of doing that I will consider these salaries on a cylinder clients simple Omega background he Vogel was excellent which is nothing more but the generalization of coal supply and reduction instead of the friendships windows so we will present the service on behalf and the claim will be that epsilon will be quantities they should Of this classical and working to rebuild system were actually equals aged In some form so by turning gone bad absolute there I will calculate spectrum of answers here that's a philosophy and I explain where it came from so next two-and-a-half hours I will be during the examples all these 4 2 particle or cases of teaching system I never said that teaching system which was on the blackboard here had anything to do with the intolerable system worked and I will start resistance if there is a classical intolerable system associated presented to this phase space from the Model a space of solutions to here and then I will consider still particular examples which will build elliptical Jerome was and periodic told that which is identical Jerome was is that of digitization of this where I played between sign square sing square and that's what you do not ask me tribal and I will answer questions like What was asking me for that so I will solve it In a sense that I will produce 1st exact formal for spectral which will be expansion in picture in the next financial planners .period sold the bombing and that the expansion will coincide with the instant on expansion here so there is a Zeisel instant greater swung open and use of correction instant on corrections will be exactly as expansion since its financial of minor size miners .period they will coincide and I will observe the distressing coincides and I will make so this is a bodily functions that I will make general statement it is always to endure I boxed equation and this will end up for some paper wrote was West which is very general and in such language so we will depart from the language of Surbiton equations will move to the language of Boxster equation so we will get to see laughter equation for the system as we which some wasn't have been OK and goes about the equations for us will be the statement that there is an order parameter in quantum field salary which is determinant of X minus a scholar field something which we have been more among and then there's that guy will satisfies awarded to that Ward identity will not be awarded entered in sense that I don't know for which transformation it is produced but it's an identity and that was allowed to determine to what Baxter equation we're dealing with is engaged so we will give it was a solution and those Doeschate salaries which will be like status will not give me any more as a floating on real solutions the will they will give me a certain analytic provide solutions in the Lancet property will be determined by asymptotic Of the boxer operators that can be calculated to engage salary and asymptotic sing all possible ways of turning gone on and turn engulfs the parameters and describes its space in FY hopefully I get to it but not is important thinkers to run this logic said we 1st get when we get there when we look at cities and with to than go to generate and then moved back to what's happening and we will realize that on the way lots of interesting questions will be answered in particular the question of superfecta Moon calls this young functions will not be answered that itself from which thinking the and


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