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Supergravity and its Hidden Hyperbolic Kac-Moody Structure­s

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right on my so it is a great pleasure of 2 speaking for the victor straightforward Victor for answering many of my naive questions I mean that a few years ago we we discovered some about the activity cats with the structure popping up in the in the gravity context and I knew nothing about the field so many times I center talked to Victor and I sent him an e-mail saying he always answer very kindly and my questions I want also to to congratulate him on the fact that he has just been elected a member of the National Academy of Sciences in America so that so
yes so I probably cats with the structure so that the talk of the topic I'm going to talk about was something which was 1st conjectural by Bell now Judea and back in situation should immediately say that it is connected to although there are cited of different ideas and I conjecture by again and also by Peter West concerning EU 11 but what we did find evidence about was that he of the which is and I probably version of Yates in the Super balloting one-dimensional and but today most of my talk actually will be a linked to a baby version its say of the 10 which is called by fixed or 3 so it's the public version of it won a case of assent to Witchazel ticking diagram of this type of which means that it has a from a tree which is something like that and minus 1 minus 1 and then 0 elsewhere so this is mostly the object that I will be talking about but why are we interested in this so we some work with firm marking all done here actually and we we're studying the general solution of Einstein's equations near a cosmological singularity in Super gravity and then we found that the had the chaotic structure which was described by the the device able to all of the 10 10 and then after that we look more closely at all the aspects of this and work with them not only market nobody ,comma Nikolai and also accent Kleinschmidt then and others and we found more and more evidence for the role of the site Beverly the cats with a certified public means that you have a couple matrix which has no signature of minus prosperous in my case so for it it will be minus and prosperous prosperous trust and we will discuss here the growing 3 examples where you have a bias might put the minus 1 in the wrong place at so it's a freebie the matrix of the minus 1 is here and then we conjecture and we've in no and Nikolai that's there was actually an equivalence duality between the Super rugby team 11 dimensions or more generally what is called and theory so what contains a super string theory and all the objects extended objects and that something which is of course set based on the 10 and this course sets is the concept of it then divided by the maximally complex but all the time and the basic duality is to say that a certain theory 11 dimension is equivalent to the quantum dynamics of amassed spinning particle on this course said space so in the
picture the ideas that on 1 side that you out of an 11 dimensional theory which means you have a certain number of when you look at the zeal molds fuels which is guaranteed to the former the gravitational and then all of these "quotation mark objects extended objects associated to that in theory and you you the idea is that the quantum dynamics of this system would be equivalent to the the quantum motion of particle particle but with some speed the ways of freedom on an infinite dimensional space which is detained divided by its maximally compact so here you have fewer new Needleman dimensions and here you have a particles so something which is moving on with 1 parameter but you quantise
it's so what was the evidence for this the evidence was that's where 1 the 1 that partial evidence on me which is you have the the field equations that safety officer provided 13 11 dimension and we could show that if you look at the feud accretion of Super rarity and you look at them near the big bang on the corner singularities you couldn't would use an expansion in which is labeled by a certain number of of integers which is the number of spatial gradients that you take on in a general rule a component of the field equations and you assume a lab Belinsky hot article for shapes and that the spatial gradients are in some sense small compared to the time ingredients so you have an infinite series labeled by all those integers and on the other side you have the the motion which is just a geophysical this infinite dimensional corsets space but you can also buy the expansion in the you know all of the generator because there's an infinite number of generators you can also introduce a height expansion of the equations of motion and then we were which are parametrized by awards and then we could show that up to level of 30 in height there was an equivalence between the 2
sides and they wear several other aspects that could be checked but there is something frustrating in this line of work like maximum mentioned the other day that in a sense this is that there are some things that work nicely you can pull some things but you never know whether they're afraid this is mathematically sordid whether there is leading agreements between the 2 things so recently referred to the extent that we decided to do different things which is to do a concrete case study which is to study 1 particular sub example of Super reality which is that the reduction of the program it is not in the 11 mentioned but in the press 1 dimension so super rarity nowhere space-time dimensions the space dimensions and one-time dimensions and we looked at the Taiex squashed CIA which means it's a cosmological model will you it wasn't the geometry of space as being not that you wrong freeze feel that would be trivial but you you deformed the sea squashing it in no way that I will explain so that it's I'm in the face fear metric he's variants on the left and as you too will open the rights issue to walk so you keep invariants and other leftist you to walk so you have a geometry which is a genius but said you can translate the geometry over the Friscia but it is not in variant on the order of the whites as you so your something which is deformed so intuitively this means you look at the geometry which looks like a white people will care and in this geometry will evolve in time together with the family degrees of freedom of exactly and we will look at it as a function of time in projected out what happened near big banks in writing and we will do that in the supersymmetric fashioning a quantum fashion quantum Minxia in the physics and so age but not security form but we have equitization the so what is technically a what we are doing is we consider the actions of simple supervisory tea over with his goal of junkie 9 cosmological model which is this extremely squashed the sphere which means space-time geometry we will this time that's just gage factor of which you want to change the unit of time and then the space geometry is just a function of time when you express it in the in function of the local frame which is Mason made therefore left in that 1 forms over as you to criminal 301 forms that satisfied this so from the joint at 1 point of view this means that I have this time actually squashed the How
do you tries the original left invariants having issued to left metric is defined by giving them the metric and therefore politic form format 1 . over the issue to manifold and then transforming it's all over a space OK so the geometries Spiner twice by a matrix of out the symmetric matrix the local geometry in 1 frame and uh but you can't ,comma tries this symmetry the May the military might of Gaza the composition so you can decompose it's in diagonal elements and then a politician matrix principles on the left and alteration matrix on the right so as his alteration matrix which depends on the relay angles so this way I have planted tries the 6 components offered the by 3 symmetric matrix by free dialog components so the debate that has allowed the call value them all of the diagonal elements of the matrix OK and the and then see you later angles which is the angles of alteration in order to put this ellipsoid type thing in diagonal for now this was
for the boss the degrees of freedom but they're also the chairman degrees of freedom which other gravity known as components and we had was guy regional components in a special gage fixing framer and in this the guy regional components they have to have 1 spin index which takes for values and they have a special index which states the values so we have 12 Convertino components and therefore I which only presented after some transformation by some from antique objects which of 2 induces an index a which takes the values and index capital in which takes full and these other components of the
in the degrees of freedom in a local friends now you take the action of Super rugby team and you you be plays this geometry need to see what is the dynamics which is predicted by supervisory tea as it is and and then after going to OK the undertone version you find this type of action where are the big categories of freedom are connected years this thing is too small so they are so this section as the form an Anatolian action so I have conjugate moment at times time derivative of call that's so the 3 components be are there for the volatiles that measure of deformed shape of the geometry and there could you get momentum of called became the defeat as that should qualify idea earlier angles of which is the orientation of this quality metrics and that could you get momentum this season the Phi Phi Delta from unique terms OK and that and then the rest of the action is multiplied by some extra components that come either from the matrix is where compliments in the metric and this was 1 of those you'll component of the gravity just by looking at these physicists will notice that these Simon and mathematicians who noted that these Allied Automotive IRAs OK which means that part of the equations of motion is that the objects C on the rights of 0 because I have 2 very by thing on so all of those things will be constrained to Brazil but TI and the QinetiQ terms is for the gravity and I have written here of aquatic forms so this climatic form which is the 1st hints that there exist and this was the first one we found that during the 1st hints that there is a cat hyperbole cats with this torture is this quality CFO this is the politic form defines a new dimension as the sum of the squares of fur an object's minus the square of the some of these and objects so this is and I public rhetoric and and and this will be the Mitrokhin cast space off the with the algebra 83 in that case a case and this metric usage becomes CIA and therefore defines the QinetiQ terms of the gravity but when you look in the action before going to the Newtonian version you find that the QinetiQ term of the of the shape of the geometry is also given by the same CoreLogic form GB times the derivative of the logarithm of the shape of so so you see and but this is linked to supersymmetry here than the same CoreLogic form is coming in the Bozanic sector and in the firm in the sector and as I said the find that job levels which apply means that you will have constructed now and you do the quantities Asian the hunters Asian consists in saying OK I have I can do simple conservation at the conjugate moment presented them intruding alike want decision by David Tua predators with President beat that same thing for the earlier angles but for the feminine degrees of freedom as I have a kinetic term which is 5 5 of the condensation is that the onto committee of the fight among themselves must be equal to that the a unit of OK but this is a key for it's OK for the relation admits I have felt objects and then the onto commit itself to these objects is essentially aquatic forming the right-hand side so this means that the gleeful about which appears in the sky in 8th we've 8 classes so signature plus minus 8 4 and therefore the the the wave function of the universe which is the quantum implementation of the dynamics of the geometry inseparability described by this model is an object which is a spin-off of this twofold underboss so it is an object which has 64 components and uh and gravity operators are not what physicists call government Beatrice's which are 64 by 64 thinks so so here we have a quantum problem but instead of having like initially I question you know one-way function you have 64 we're functions cobbled together and then there is and what is the shredding their like equation showing like equation to come from the constraints they said that all the operators that the with liable for enterprises in the action of 2 vanished classically which in the direct sense means that when they are applied on the state's so he upside denotes that the wave function of the universe so it is a Southern states and as a step and operator also hear you're right that the action of operator which is a differential operator riding on this subject is the so here you have 0 and unions these constraints you can solve exactly because you find that there are equivalent to saying that it would function does not depend on the earlier angles so it simplifies very much you have the wave function depends on the finally on the shape of the universe how it is deformed but not of anger factors and at this stage the eh equations of motion of this quarter model is that the wave function which is 64 complements depending on the valuables must satisfy these equations the questions here and 1 question not for today because he decided by is a differential operator also this these partial differential equation in the rivals but I'll many beauties do you have here you have a full-time 64 equations here and another 64 accretions to fall only 64 on loans so this is a heavily over over-determined system of India's which because of supersymmetry must be consistent with the consideration is not anomalous but this depends on computing the commuted the supervisory board of the Of these operators and and this is where we did explicit computations like these operators which the supersymmetry constraints they have this form they have leading term which is linear implies which means the impartial delivered debate the better multiplied by the gamma matrix so it's like a Dirac equation each 1 for fixing dates in the system would be the Dirac equation of but then you have more complicated terms including terms which are cu In the from and so you have like a cubic there are 2 questions I don't recently from Viktor Orban said but this is what people stood in the name of the Cuban director Christian is a simple things that than these objects here so so this is the supersymmetry constraints and in its U you already certain objects appear like appears twice bitter 1 here and in these objects here they appear in the Code tangent of beta one-liners better 2 which is this which will turn out to be a waltz OK so there would start appearing here of this gets model under brought up this is the true victims so this is the 1st time in this business that people kept explicitly although not enough terms in from its CU in the supersymmetry genitals and quality in the action and in the Amazonian so we have checked explicitly and it took us some time to find the sensor that's the onto commutator all of these objects clause on on on each of you he won't let me remind you that we want to certify that the vanish on the state's so therefore the on the committal of to should also vanish module as all H which is another constraint and indeed this equation is compatible with this and therefore you have an open so open means that on the right hand side I don't have concerns about concerns but also functions will be stopped but
it looks good and so the main question we wanted to address in this where is out there really hidden cats with these structures in this thing and and we can give a precise answer this is the full and so for the amused tone in the tone in other parts which is quality can the pies so this spot means essentially a kind although integration so it's the denomination in Iowa three-dimensional space because I remind you that this duty is a Gaborik metrical carried the signature of minus press press so this is like a applying Golden operator and then there are potential terms and also a master of now if you look at this potential terms function of data they have an explicit form and there is no doubt government of several parts the simplest spot which can you can see if you are not in supra gravity already is a sum of exponential offering forms to beat us so this is like a told that potential and then you can often distort the potential walk out the door ,comma linear combination of the because that appear here and this combinations of precisely the 6 level 1 the wallets of 83 when you decompose with respect to anyone so if you take a the composition of the subject but with respect to the A-1 which is just which is what which is just this uh and then you find that you can decompose the infinite dimensional 83 caps with the cats with the underwriting an infinite number of levels at a level of 0 you get what you get simple would switch out the differences b . 1 woman been that to bitter to minors of and b Tuffey buying been 1 which other woods associated through the GAO abroad of 83 OK so sorry yes we don't pickup was where is it we want you to know you have inside this summer to remind of the summary of GM-free so you decompose 83 respect would you think of this as healthy but also due so these are the words of the GMO-free 100 by the end of part of the Italian contains also know all the exponential so this contains no other Woods which is of how preserve file 1 and I'm in a bit of money has become 1 and become 1 was that to that appear before which shall now although Woods at at the level 1 and they come with a certain quantum of potatoes called J. here J. here and here you have also all the terms in 1 over seems square of this would say at which comes with the square of another operator which is written on the next thing so these are explicit calculations OK you you take a physical modem you quantise its use of the Internet makes and then you find that all of those quantum operators the key of the enter the is a serpent under bike function is supporting in these things and now your remark that the 6 operators also is 1 2 with 2 3 is the 1 G 1 1 due to 250 they don't generate exactly like commutator was a 60 four-dimensional presentation of the infinite dimensional maximally complex of proper K phase which is the complex about about formally defined as the fixed point of the I'm in a fixed set of the show violating pollution so as Maxim as reminded us and you have to be here age and there is the shoveling pollution which changes the minus F etc. and then if you looked at the fixed sites although this evolution this is the sad generated by the EIA miners and fight which still generates an infinite dimensional and and this operators they do satisfy exactly the commutation relations offered the equivalent minus and fight EFI defined by this recovery with the algebra and its compact but and this is a nontrivial thing because to do it you see you have all those complicated things that come from superiority and these operators they satisfy the quantum of onto commutation relations which is this fold under what type thing so you have to prove that the commutator also enters the country details of these things is actually the committed of of the spinnaker tells like this 1 is to use they but 2 of the committed tells of these Jaber precisely reproduce what we farm before which is the existence of a finite dimensional spin oil representation of K in the isn't the real thing so this is an and then so this is a very precise statements about the presence of operators that do generates K of at the and but there are the most complicated terms in the calculations of the terms "quotation mark seeking the from cocaine which were known to be complicated things and that the the terms "quotation mark the fermions finally they separated in the transaction before some of them were problematic and somewhere or with aquatic and deferments and then all the terms are written before and told like potential told means exponential of somebody now forming the bitter now if I am the the the the forms on file be done as our linear equations in the Times OK so they defined I proclaimed so these are the waters of the site that would in these other divide shambles this album the walls of the vise chairman's and all the other chambers and if you want to see very far from all those walls on this exponential would be small and then they really meant something in addition to distort I like things you have this new script which was the most complicated thing to computer and this new Square you find remarkably although it's very complicated initially that's actually commutes with everything else in the Germont it belongs to the center of of of this 101 generated by S & J and not only this but it is itself when you constant the square offer very simple operator OK which is this 1 which is erratic now in the in the firm so you have very big simplifications coming in and and then you can discuss explicitly that the solutions for the quantum universe describing supra balloting single now you need to solve this do act like equations so you you look for 64 components :colon vector of the functions of the Bible and they have to satisfy full-time 64 immigration so that we could discuss in detail what is the set of solutions of these things depending on where you are in in quantum space and actually the various quantum states there are nicely Palmer tries by this operator because this operators like a film number operators things value for minus-3 to Presley and there exist solutions 6 the number of of that from inappropriate
and in particular but you find that the full sum From unique levels you have only discrete states you have exact solutions like the kind of bond states and excited states which depend on you own the small number of constants and then in the middle of from experience you start having functional freedom in the sense that the general solution of this equation is parametrized by free foot the functions of the Tory functions of 2 viable and then you can go beyond that and say OK let's consider in particular the wave function of the University of Illinois in the effeminate levels where it contains functions so this way functions use can be made of the army can look like a wave propagating in internal space of the universe and this wave is submitted to the UN in it's it's it's needs these wars of the war's connected with the simple Woods of the vise chairman for 83 and in some approximation which is a W K B approximation you can ask when I have this 64 component we of the universe bouncing on these walls what is the connection between the states after reflection and the state before friction so my affect almost 64 components is easily now victor in the representation space 60 four-dimensional implantation space of cable 83 of this Compaq cats with the algebra and and what you find in confirmation of what we have fun with him in some cost money and approximation is that in the W K B limits the states of the of I'm in the state of the universe after reflection is given by applying on the previous state of September reflection operator but this reflection operator who lives in this representation of the spin oil viable and therefore it's in fact defiance of marginalization of the Bible which is not a boss on the type of I hope in the viable Puget your of reflection operators in the square for reflection Labrador is is 1 by definition of reflection on the wall but here there's reflections of our eight-foot of unity you need you need to take the 8th power of the reflection 12 unity so because it is a spin on your extension of violent reflection and you'll find that some of these general nice formula that reflection of upright on any of these awards simple rewards is given by exponential minus side by over to the same jailbreak tells that appear OK so although those operators which are 64 the mental implementation they also define a spin or extension of the Bible which is present definitely in the quantum dynamics of the universe therefore let me and conclude so this is 1 particular case study but it's 1 in which was really good to completely into account the quantum aspects of the dynamics of fertile actually squashed the sphere uh in Super Reality and this study definitely confirms that the hidden presence of hyperbole cats with structures in the sense that the undertone is made of building blocks and each building block is a operator which is presentation of the trip was announced the same relations it satisfied the said relations for the object's CIA minus if fight already I should say this all the operators that appear here they wear the representations of these differences can you had the EIA fight HIV you at the level of the wood so we could see on now something to this but finally need them In the end Newtonian the operator that appealed where the presentation of the minus F which is the Compaq but in the same way that if you have the G-11 and if you take the anti-Semitic mattresses you have your complex of apology Jilin and so this is the analog for and so we could discuss the the wave function of the universe and pools were income told the number 0 solution of these things see that the square must terms which is "quotation mark again the pheromones belongs to the center of knowledge which is the 1 generating the combat part of free so we hope that this result will help in clarifying what was towing the conjecture we women before about it and because here we could do all of this calculation for the same under but I should say um additional statement that part of the calculations of the explicitly with 64 by 64 mattresses because some of these calculations on difficult to Dubai and so we use some of the computer means to compute publics of 64 by 64 minutes which to check everything and in the case of the attended dimension of this mattresses would be to to the 320 so which means like 10 to the 50 so it would be impossible to the explicit calculations but probably from this very simple results we have here 1 can guess what would be the case with a structure for the full of superiority 11 dimensions which is the real challenge OK thank you for your attention fewer but the official who deals with the Canadian also the reflections of 1st air if you look at quantum states that decay exponentially you you have modifications of this fall melodic for instance in some cases their reflection if 1 is isolated from the other 1 the explicit quantum effects of the reflection is given by a crewmember function so you haven't confronted pugilistic function and then you have a defacing upon reflection which is more complicated by Justice this thing but it's a well-defined affection for the formation of who is out there I don't know if 1 should pose the question in disturbs In an interview Newtonian because the Tony and is still a limit on to constructed from the operators that do satisfy exactly the usual I'm in the same type of cats the relations OK and so the fact that you have quantum effects that modified abuse can be with the result is not a big deal but your right question is meaningful question regression the dual picture that was started by a end unit of of the article for that was already in this example come here this year exactly so uh so in a
sense of what we have found before where parts of
the picture that we have here where the evidence we have before could be seen it was just like at the buzz quarrel can fall when we look at the instruments in the gas money and approximation that this was this copying in the Newtonian which was part of the Soviet farm possibly but here we wanted to still get what is at the end of the level of quota monitoring and without approximation and also the Caltech terms from because if the terms "quotation mark again from inside most of the things but the fact that the commute exactly in fact I want to ask this question now that victory is back with us we formally this new squared terms the master and should be you Kazimiyah of resale and formally this chasm is given by an infinite son of 1 of the genital Square which is not well defined I mean you could define the Kazemi a full cats Moody because it would all the things but here we have compact cats with the where a it's not it's not because the thing you cannot know more order of things OK so they aren't question but the fact that it commutes like Academy is a very good portions of it ,comma origin Washington history of the region what do you think was screened year alone central after all it so it's you it's an issue to invariant so it's emotional issue to it was generally assumed to invariant metric most general so it as 6 parameters of free shape parameter of "quotation mark questions but this question directions you operations well actually thinking a few and
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Title Supergravity and its Hidden Hyperbolic Kac-Moody Structure­s
Title of Series Après-midi en l'honneur de Victor KAC
Number of Parts 6
Author Damour, Thibault
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 10.5446/17002
Publisher Institut des Hautes Études Scientifiques (IHÉS)
Release Date 2013
Language English

Content Metadata

Subject Area Mathematics

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AV-Portal 3.5.0 (cb7a58240982536f976b3fae0db2d7d34ae7e46b)

Timings

  671 ms - page object