 Jag vill tacka alla organismer. Det är bra att vara här i ITS och för att sätta ihop en liten grupp av människor. Jag skulle också apologera på två siffror. En är att det har varit många år sedan jag jobbade aktivt på den här siffrorna. Min människa är kanske inte så fräst som det skulle ha varit annorlunda. Den andra är att jag hoppade att få den senaste veckan för att bli lite mer intressant. Men jag hade en kalld som du kan lära mig. Jag planar att fokusera mer på den spektralen av de människa människa modeller. Jag ska diskutera varför det är såklart klassiskt. Det är sida, kontum och de supersymmetriska människa modeller som kommer från människa. Planen är att ge lite mer av introduktionen. Vi har hört att många av människa människa fokuserar på att lära sig till att fokusera på de olika människa modellerna. Men det är inte svårt att få ut några extra saker. I den här videon har jag sett att många av de som inte är expert i den här människa människa fokuserar på det här. Det kan vara bra att förstå det i en mer detalj. Och fokus på spektrum och grannstadig konjunktur. Det är en avsäkterning om det är den sista människa människa. Det är en avsäkterning om att lära sig till att lära sig till den positiva människa människa. Det är en avsäkterning om att lära sig till att lära sig till den positiva människa människa. Och så ska vi diskutera varje avsäkter till grannstadig konjunktur. Det har varit många år sedan och det har varit en avsäkterning i några av dessa. Om vi recall, vi är considera en bosonic member. Vi börjar med bosonic case. Det innebär att vi har en invading av en människa människa. En världvolum i spacetime. Det är en avsäkterning om att lära sig till den positiva människa. Det är en avsäkterning om att lära sig till den positiva människa. Vi har en vänniska människa som är svarande i världvolumen. Sigma är fixat. Det är en avsäkterning om att lära sig till den kontaktiga människa. Vi har invading av ordentliga funktionser. Här arbetar jag i den liten människa människa människa. Även om invading i spacetime är r. 1 plus liten d är parametret i de här modellerna. Och en vänniska människa. D är essentiellt de transversal människa till människa. Vi har d är en avsäkterning av ordentliga funktionser. En avsäkterning av ordentliga funktionser. Det är en avsäkterning om att lära sig till den människa människa. Det är efter att lära sig till den liten människa. Och lära sig till den människa människa. Den här energi är byggad av Hamiltonian. Funktionen på spacetime. Vi har de x-kornet funktionser. Men också de konjugerade momenta P. Så det här är Daniels integral. Den här fixade särskilt. Vi ska tänka lite om det här. Vi har de momenta square. Och så finns det en Poissonbracket. Vi har Poissonbracket av ordentliga funktionser square. Det är en avsäkterning av ordentliga funktionser. Vi har två Poissonstrukturer här. En är givet av siffrorna. Här har jag sett den här normaliseringen. Men det finns fläktighet där. Det är en av de Poissonstrukturer som vi har. Det är en avsäkterning av ordentliga funktionser. Men också de dynamiska Poissonbracket. De här ordentliga funktionser och momenta ska vara kanonisk eller konjugerad. Det är en Poissonbracket här. Det har varit normaliserat. Och jag kan säga också att en av de detaljer som kräver den dimensionella reduktionen från den fulla teorien till den här siffrorna. Man måste också få ut några konstrainer och så vidare. Det är en diskussion i en paper med De Woll och Hoppe. Vi går över till den matrixkontexten. Vi har sett att de ska replacea de Poissonalibrar av funktionser på den här siffrorna. De har också varit normaliserade. Vi har sett att de är zero-min realvaliga funktionser. Vi representerar den Poissonalibrarna i terms av matrices. Vi går över till en algebra av traceless Hermitian- och biomedmatrices. Tracelessnessen är från zero-min och Hermitian- från de realvaliga funktionser. Det är bara n, det är matrixsize. I den här representationen representerar vi Poissonbracket- med konvitatorn av matrices. De intervjuar på surface och går över till tracelessnessen. Det är en fin fiktion att det finns en konvergen- så att vi kan, lite independentligt- av det aktiella topologin och så vidare- vi kan arrangera basis av traceless Hermitian- matrices, så att strukturekonstans- i den här basis är konvergen. Om du vill använda högre genus- surface för att parametra människa- hur kommer det att se upp på den visst sidan? Jag tror att du behöver skriva olika basis- som ger dig strukturekonstans. Jag tror att Jens är det mest... Det vi diskussade tidigare... Detta ansvar är den här. Det finns en abstract theorem. Många olika saker. En, för exempel, Bordermann Meinröngen- som provar att för någon genus- surface, den matrixalgebra- kan vara konverterat till den genus- surface. Men det vet inte vad basis- för högre genus- surfaces är. Detta matrices, i vilken du får- strukturekonstans- och A, B, C- är bara för Torai och Sphere. Så det finns en existens- av basis- och en strukturekonstans- i basis- så att du får konverterat. Det är lite mer trevligt. Men hela idéen- är att det respekterar- den här- repräsentationen. Det respekterar symmetrar- eller av konstrainer. Vi hade- i den originella- människa- difimorphism- invariant- på surface- och en- area-preserving- difimorphism. Vi vill- konserva den- area. Det här- i den matrixen- modellen- representerar- SUN- invariant. Så den- speciellt uniteria- för den här- area- konserva- konstrainer. Så- så vi har- den- Hamiltonian- som vi har sett. Det är nu en trace- över- momentum- sidorna- matrices- square- och sen har du- komiteter- square. Över- all- the pairs- of the matrices. Och- if you want to write this down- in terms of a basis- of SUN- and some standard basis- then you have just the sum of- and then we have- we have- the- the- structure constants- and then- so this will be the- the- the square. And then still- we have- we still have- so the canonical Poisson brackets- between the coordinate- and momentum variables- here. But now there's only- finitely many of them. So that's the whole idea- here that if- since we have a finite- dimensional system- we can then- quantize this- in the standard way- using the Schrodinger quantization. So- so we represent this- and set of- coordinates and momentum- on the square integrable functions- over the corresponding dimension. So we had the d- space dimensions and then- this is the matrix dimension. The basis- the number of elements of a basis- of this space. And then we represent- our Xj- as just multiplication operators- on this space. So we have expanded- in this basis ta. And while the momenta is represented- in terms of derivatives. So minus i times derivative- with respect to Xj. In order to- to represent the- the canonical commutation relations. So then our Hamiltonian that we had- the sum of p-squares- goes over to the Laplacian- acting on this space. And then we have a- scalar potential essentially. Which depends on the coordinates. And I will come back to- the details on this potential. But what we have here is also- symmetries. So we have- essentially the d- dimensional space. And the rotations- maybe represented- within this. Essentially Hamiltonian commutes- with such rotations. But it also commutes with- rotations in the- in the matrix space. Also SUN transformations. Which are then represented if you- in this basis by- orthogonal transformations as well. So what we have to remember is that we had- we have these constraints- of the thermomorphism invariance. So we still have that constraint- which we have to implement. So actually the physical Hilbert space- is not the whole space here- but rather this where I wrote- the sort of bosonic Hilbert space- physical. Which is then the SUN invariant states. So the SUN- so the generators of the SUN symmetry- can be written in terms of this basis- with the structure constant like this. And that acting on our states- should be zero. So that's how we implement constraints. And this is just amounts to the standard- direct constraint quantization procedure- that we still have some constraints. So we will look for- to solve those constraints on the quantum side. This is the constraint? This one is a constraint. So this is- at this point. I don't remember. Yeah. It could be second class at this point. So yeah, they have to- this is probably written in this paper. Okay. So- then we have this complication of- supersymmetry if you want to consider- the supersymmetric membrane. So the idea is to somehow add- spin degrees of freedom- and obtain a supersymmetric theory. So I will say something briefly- about supersymmetric quantum mechanics. Because it's not the full- supersymmetric quantum field theory here- but it's really a reduced thing- into the quantum mechanics. So then it can be more simply stated. So essentially a supersymmetric quantum mechanics- you can think of having a couple of different objects. So one thing is the- there is a Hilbert space. There is a grading operator K. There is a Hamiltonian. And then there is a supercharges- supercharges operators. So you have a Hilbert space. The grading operator is just an operator- upon the operator is going to 1. So you can sort of split the Hilbert space- into a positive and negative part. Or the eigen spaces of this. And you can term this- the even sector and the odd sector. And then this Hamiltonian operator. It should be even with respect to this grading. So it should map even to even. And odd to odd. And it should be a self-adjoint operator. But then you have the supercharger operators QJ. And if there are N- the curly N of them- that we can say that there is an N extended supersymmetry. These should be odd operators. So they are mapping the even to the odd. And the odd to the even. So they are interchanging these spaces. And they should satisfy- essentially the Clifford type algebra. That the square of QJ- should be the Hamiltonian. But they should also anti-commute. So this is the definition- of the supersymmetric quantum mechanics. And some nice properties that are implied- just by the structure is that- your Hamiltonian will necessarily be- non-negative operator. Because it's the square of QJ. I should say these are- in this formulation these are self-adjoint operators also. So the spectrum of the Hamiltonian- has to be on the positive real line. Then there is a pairing between- the eigenstates of the Hamiltonian. So if you actually have an eigenstate- with a positive energy of the Hamiltonian. So say that you have it- and you can use this splitting. So you can say that you have an even state- for instance, with positive energy. Then by just acting on it- with the supercharge we get to the odd sector. So you would have an eigenstate- with the same eigenvalue E. On that sector. So there is always a pairing- for the higher energy eigenvalues. But not necessarily on the zero energy sector. Because then you would annihilate the state- by acting on Q. So this is used for instance- in index theorems and so on. To spot whether there is a zero energy state or not. Okay, and then in this- in the supersymmetric formulation. What we want to do is somehow to- instead of having SOD- as the rotation group symmetry. We want to have a spin D- represented on trivially in our space. So we are considering actually- representations both- sort of spin representations- with respect to the RD space. But also with respect to the matrix space. And these should be represented- as some bounded- operators on some- folk space. The Hilbert space of the model. So we then have to go to Clifford Alibas- to construct such representations. So starting with the D- dimension space. And we're looking at corresponding Clifford Alibas- over D-dimensions. So say gamma matrices- satisfying the anti- computation relations. And then you take an irreducible representation- for these matrices. Or just- a certain matrix size. And I call this ND. So this depends on the dimension. And there is a certain natural- irreducible representation here. Then you construct- over this space that you got. You then sort of couple that- to these matrices. So the matrix degrees of freedom. And then you construct- a Clifford Alibas- on top of that space. And then you stop- in dimension a bit more. And then you consider the irreducible- representations of that algebra. And that will be even bigger. So then it's- the dimension is 2 to the- and then it's essentially this- dimension here. And D times N squared minus 1. And then half of that. And we will see explicitly how you construct- these things. In one way. Ja, så här- we consider actually a complex representation. But you can ask- whether you should want a real or a complex one. But typically- at this. So here it's the first stage. So it depends a bit- what kind of structure we can get. So I will come to this soon. So this real representation- in certain dimensions this turns out to be- actually a complex or a quaternionic representation. And then you can use that. So- but on this stage- you can also ask- should you have a real or not. But let's think of just a complex case. Otherwise it would be more complicated here. So- what happens in this picture is that we- so adding- to our bosonic Hamiltonian. So the- the kinetic energy is actually plus the- the scalar potential. We add this- so spin degrees of freedom or this- fermionic operators. The theta. And they come in with a- linearly in the coordinates- X. The rest is just this sort of- structural constants. And then it turns out that you can- write down a set of supercharges. So the number- of supersymmetries essentially is this- dimension here. And so you have a- linear in momenta. And then a- quadratic in the- axis. And then coupled to these- from variables. And then it turns out that- for certain dimensions these- satisfy this- supersymmetry algebra. That- the anticommutator of such- supercharges gives you the Hamiltonian. The physical Hilbert space. So this is just the- generators of SUM. So on the physical Hilbert space this- piece vanishes. So you have- it closes up to this- supersymmetry algebra. So in order to understand this- requirement. One needs to know a little bit more about- Clifford algebra and so on. But I don't have so much time to get to it. But let me ask- I. And then you consider the corresponding- Clifford algebra. And these are well known to be just- essentially matrix algebas or- a sum of two matrix algebas. But the matrices are- have a real or complex or- quaternionic structure. And so- this is just the dimension of- of the Euclidean space that we- construct this Clifford algebra over. And then this is the dimension of the- responsible representation of the Clifford algebra. And this you can read off from- this side. Then I have- essentially just written what- sort of the structure is in this- in these spaces whether there is an- additional complex or quaternionic structure. And then on the right- hand side here is what happens when you act- essentially how can- spin D be represented on these- spaces and then there is an- additional splitting into- either you have just- essentially an irreducible- representation or it splits into two. Either different or the same- representations. So actually you can see that the- special dimensions where you have these- supersymmetries, the case. So for my- label of D here it's the two, three, five and nine- cases. This is what I've marked with the arrows here. And it turns out so in the two- case essentially the structure is real, but in the three case the structure is complex and you can use that. And also in the five-dimensional case there is a quaternionic structure which you also can use. While in the highest dimensional case which is of the most interest is essentially a real structure but it turns out that there is sort of octonionic features in this case also. I have also- indicated this one here which is somehow a degenerate case, the one-dimensional case. So this is somehow- if you want some kind of zero- dimensional thing. So this is again related to the fact that you have- essentially the norm- division alibas are the real complex quaternionic and octonionic ones. And this is sort of the same thing which appears here which has to do with Clifford alibas. Okay, so I might come back to this- table. So now our full Hilbert space is then the bosonic one times the fermionic fox space where we have represented this Clifford alibas. And then again there is a physical Hilbert space which where these constraints vanish but now it's not only that bosonic sector but there is also the fermionic part here. So that's imposed as a constraint that we should be in the kernel of these operators but then you still have also a symmetry sort of a rotation- spin-d symmetry which is then essentially a rotation among the coordinates and then also rotation of these Clifford alibas or on the fermionic fox space. So what we would like to emphasize here is also how we can construct these fox space representations. So essentially the one way which was used by David Hoppe and Nikolaj is to essentially you pair up so given these stitas Clifford alibas you can sort of pair them up to construct creation and annihilation operators. So one way is just okay you take the first half and the second half and pair those up and then you have essentially creation and annihilation fermionic operators satisfying canonical alibas like this and what has been done essentially is to to split the space. So along with this pairing you use a splitting of the d-dimensional space into d-minus two variables and then the last two you use as a sort of complex variable you can take the real and the imaginary part of that. So I have here written X prime is the first set of variables here and then Z is the pairing of the two last variables d-minus one and d and if you do that this sort of goes well together and you can write down the Hamiltonian in terms of these creation and annihilation operators which involves these first variables here and so big gamma is the reduction of these gamma matrices in terms of this splitting and then you have the last part so this first part preserves the fermion number but in the last two parts you have a raising of two or lowering of two fermions fermions so in this sense the whole operator result of mix the fermionic sectors but there is this alternative in three and five dimensions which I'm not sure how much it has been discussed, okay it's also been used by Clausson and Halpern and then discussed this a little bit in my thesis that in these special cases since you have this complex or cotonium structure you can use that pairing which gives you a more canonical pairing of the fermions so that in that case your Hamiltonian can be written so using these other pairing you have essentially yeah so your Hamiltonian is then the bosonic and then you have a piece which conserves the fermion number and then the SUN and the bin D generators also sort of preserving fermion number then there was this degenerate model which essentially so it's one dimensional case which essentially is just a free Laplacian acting on the on the matrix degrees of freedom and then there's a piece which vanishes on the physical hyperspace so one way to the corresponding super charge so there's only one super charge in this case it's essentially just a theta times derivative so this is like a Dirac operator and its square is this Hamiltonian on the physical hyperspace but there's also here an alternative so either you sort of work with this or you construct and what you can call somehow a comological version which is so there's a way to also introduce fermionic creation and relation operators and then you have a Q which is not self-adjoint but you also have a Q star such that they sort of close up this supersymmetry algebra let me not spend too much on that so now we come to the question of what about the the spectra of this model so we have the classical side and just the membrane or its regularized version and then we have the quantum regularized membrane and the quantum super membrane and and one way to understand this differences is using toy models which has been very fruitful so if we start with the classical model we have this Hamiltonian again trace of this momenta and then the potential which we should note is a non-negative potential here and a toy model that you can keep in mind see if I maybe you can here so essentially we have a toy model potential which only depends on two variables so it's in R2 and it's x squared, y squared so the feature here is that there's certain flat directions so if we think about this toy model we have this in the xy plane then along the coordinate axis the potential vanishes but then it somehow increases rapidly in these cases so this is indeed the picture that arises here that in this matrix model potential you have vanishing directions where all of these matrices are commuting so it's essentially such a asymptotic direction so you can really go out to infinity while the potential vanishes but the important feature is that while you move in such a potential valley the valley also gets steeper in this in the transverse direction so because of this narrowing it turns out that you can actually there's a difference on the quantum side but if you just consider moving with a fixed energy in this potential well then you can indeed escape in the well to infinity so in this sense it's an assigned potential so on the quantum side so we are then considering this floating in operator essentially a little aplassian here plus this scalar potential and then again it's both of these operators non-negative but the important feature is this narrowing of the potential valleys here so we can again consider this toy model Hamiltonian so on the quantum side here so this is the bosonic toy model which is then just you know the Laplassian on the two dimensions plus this this potential and here you can use essentially this illustration but it's a little bit more complicated on the matrix side but essentially the same kind of computation that so considering this operator you use what is sort of happening along the valleys here to bound this operator from below by something which has a discrete spectrum so what you can do is sort of take you take half of your momenta here and you sort of put and you split your potential into two pieces and then you take also half of the x momentum put it here, half of the y momentum put it there and then you consider these two parts here separately so this if you recognize it's just like a harmonic oscillator in the x variable with a frequency y here so this is just bounded from below by the sort of zero zero point energy which is then the frequency y and the same thing here just symmetric in x and y so here we have somehow used that in such a valley there will be some kind of zero point energy due to the narrowing of this valley here which then depends how far out you are in the valley so so bounding from below with these parts you get in total an operator which has a potential which goes to infinity in all directions so because of this it has compact resolvent so it has a discrete spectrum and also it cannot have a zero you can see this for instance by sobel of inequality so this is the interesting feature here that we on the quantum side we get actually a discrete spectrum for this operator the same thing goes through as I mentioned with this more complicated potential so essentially you can parameterize this in a smart way but then everything changes on the supersymmetric side so we had this bosonic operator which we now know has a discrete spectrum but then what happened here was that we added this essentially a matrix piece here it's linear in the coordinates but then there is this matrix here also and what we already know is due to the supersymmetric that this operator is the square of something of some self-adjoint operator so we know already that even though we added this part which is not definite it still is a non-negative operator and actually it's really important so for the supersymmetric it turns out that you will have a matching between you cannot just change any of these terms with some coupling constant or something because it has to match up there is also in this case a toy model so essentially it's this bosonic toy model with an additional piece so maybe it's useful to write it down so this is the supersymmetric toy model it's essentially the same thing of the R2 and then there is the identity matrix here and then you have a part which you can write so this depends on your taste but one way to write it down is in terms of two Pauli matrices so this operator is acting on so it's L2 on R2 just in C2 so you have a the Pauli matrices acting here sort of confused why you say minus okay exactly so I'm looking at this piece here which is sort of this is the one which is linear in the coordinates times the matrix and if I square that so the square of this part plus y squared due to the Pauli matrices matrices so that means that this operator is bounded from below by at least the negative square root of that so this is what you can use on this side then that whatever this okay it can become arbitrarily negative but it essentially if you go along a potential value so if if y is zero and you go along x here then you have just minus the absolute value of x along this value and this matches up exactly with this corresponding harmonic oscillator zero point energy so what happens is that there is this exact balancing out between the zero point energy of that oscillator and this fermionic operator here so it turns out now that this changes the spectrum again so it's no longer discrete spectrum but it's rather continuous spectrum on the positive real line for this operator of course depends on which operator we consider but this toy model has the same feature features as the real matrix models so it turns out that also in the full matrix models you can prove that this corresponding operator has a essential spectrum from zero to plus infinity so what you do is essentially to use this fact that so on the fermionic side yourself in where this is as small as it can be and in the bosonic side you take a state such that this is the smallest it can be and that there will be this balance between these and essentially you can construct then a sequence of states which sort of get pushed out into this valley and in this sense a sequence of while sequence of states so you can prove that there exists a sequence of smooth and rapidly decaying states I think maybe you can even maybe not compactly support well you can take them compactly supported and normalised such that for any lambda greater than equal to zero the Hamiltonian minus that states goes to zero so this means that that this point lambda is in the spectrum The theorem is for the full this is for the full exactly so this was in David Lyche and Nikolaj where they also used this toy model to illustrate the point so in the toy model case as I said this was size should correspond to putting yourself in the lowest energy here and then this phi is essentially the eigenstate in the transverse direction and then size is used to as a cutoff to move into the valley why does it mean that the membrane is unstable so in some sense it means that it costs zero energy to just to form everything you can move into this valley so in that sense so what it means in this valley if I understand correctly somehow you can form spikes of your membrane and so these spikes somehow you cannot see somehow the uncertainty principle is not there to stop you from such spikes forming so there is somehow som ionic degrees of freedom there is an extra it allows you to somehow fluctuate with spikes but I think in some interpretations then this is used as an advantage that you can somehow if you have a membrane it can somehow form a small tube or a spike and then it can maybe form a new membrane in some other section tube somehow you are allowed to change the number of membranes and these things that there is some kind of second quantized interpretation there that it can fluctuate in the components and topology and everything ok so then there was this conjecture concerning the existence of ground state so we should think about the case now that so concerning the spectrum of our operator so we know that in all of these cases the supersymmetric cases you have a spectrum which is on the positive real axis so this is the spectrum of the supersymmetric Hamiltonian and the question is really what about the point zero in the spectrum somehow so whether there is some stability in this sense that you would have a lowest energy state where everything can somehow boil down to or if it's just if there is no normalizable ground state there that somehow tells you that things are leaking out to infinity no stability so the conjecture was that in the highest dimensional case there should be a normalizable zero in the ground state so this point should be an eigenvalue and also that there is a uniqueness there so that there is only one such state while in the lower dimensional models there should be normalizable seriös state for any matrix size and also this statement was for all matrix size by the way you separated the center of mass motion the thing is yes so this is now so if I have a k square called zero ground state I would not see I mean if this is a master's ground state it cannot be I cannot go to the center of mass frame is it excluded by the formalism no I think that's just but I think that problem probably trivializes the center of mass problem is somehow you have somehow separated these things so I think this on the center of mass side this is probably just like a particle gauges for the string when one quantizes a string if you do it badly you eliminate the massless concept ok but yes so so this conjecture is then supported by various evidence so in the case the lowest the simplest case among the full matrix models so the two dimensional and two by two matrices there is a proof by contradiction due to your relation and the proof that there does not exist that there does not exist such a ground state and if I understand this correctly it's again has to do with some of the size or that we will come to this but there is sort of the geometry involved here whether there is you know whether these states are essentially decaying fast enough at infinity and this question was then addressed I guess it was discussed somehow in the Halpen & Schwartz but then Frölich, Graf, Hassler, Hoppe and Jao considered the asymptotics in the also in the n equals two case essentially studying what happens along these valleys and looking at the the equation I think that this is Serenius state with respect to the supercharge you can then see essentially what are the decay properties of such a state how is it decaying as you go out to infinity so the question is is it decaying fast enough to be normalisable or does it sort of maybe it does not even decay it could also blow up or stay constant and then you have to be you have to know also the size of these valleys and so on so there is a lot of interesting geometry then I guess Pillen G will tell us more about the Witten index approach however I should point out that in these models because of this spectrum being continuous and various difficulties of non non-compactness here one has to be careful when you make the index computations so you cannot just assume that you have a discrete spectrum and work on that so what I would like to point out though is the existence of embedded eigenvalues actually in the spectrum so it turns out using this the way we can define the model choosing our fermions so in the 3 and 5 case shows the fermions the way we did in this slide so since we could write down the Hamiltonian where the fermion number is preserved then we can work on a sector where essentially you have zero fermions so on that sector this Hamiltonian just reduces to the bosonic Hamiltonian so in these cases due to the existence of this complex structure you can actually just split so you use this structure to somehow split the Hilbert space into different sectors of fermion numbers and on one of those sectors the Hamiltonian is just the ordinary bosonic Hamiltonian and we know that the bosonic Hamiltonian has a discrete spectrum so in these cases there indeed exist points in the spectrum which come from the bosonic Hamiltonian so that proves that there exist embedded eigenvalues in those models but this was in these cases the three in five so the problem is in the in the say two and nine dimensional case we do not have this canonical structure and indeed essentially the best we can do is to have these non fermion number conserving terms which then mess up the whole picture so it's difficult to see whether there is such a reduction in that case I have to point out here that we can on the right focus space sector we have the same Hamiltonian here so another thing which is important to point out is in the three dimensional case you do have you can write down states which are zero energy states but they are certainly not normalizable so essentially what you can use in this case that the super charges essentially a direct operator but then conjugated with with a super potential which is of a cubic type so in the three dimensional case you have also a more canonical pairing up between all these volume form in the three dimensions taking these three matrices and so you use the anti symmetric tensor in three dimensions and then the structure constants forming a real valued function so indeed this also has some properties like certain valleys where it's zero certain directions where it's actually blowing up positively and then blowing up on the negative side so it's indefinite but if you just write formally a state where you have essentially zero fermions and then you multiply by this function then this is it's a smooth function and it's in the kernel of the super charge but it's not in the Hilbert space because of the indefiniteness of this and then you have a similar state if you just switch the sign and you take the full fermionic thing so in this due to the structure there exists such states but they are certainly not normalizable and they cannot be made normalizable just by changing you have to really change the the inner product of your Hilbert space a lot having exponential decay in order to accommodate such states and also in the one dimensional case there is essentially this you can think of as a plane wave model supercharge is just a derivative times a fermionic variable so again you can just take states which are constant with respect to that so so either you take zero fermions or full number of fermions and then acting with the derivative it's just zero but this is again a constant so it's not in the Hilbert space but it somehow is better than this case because then it's constant so if you just change your you inner product a little bit you can somehow accommodate for it so this is an interesting question whether one should somehow change this the normalization and play around with whether these types of states should be included or not okay so I will not go into so much of the detail on these different approaches but essentially one approaches by construction so you write down recursive equations on the different sort of expanding so one approaches to expand essentially Taylor expand the ground state around the point x equals zero and see how you can relate the higher order terms to each other so this has been sort of fruitful that we can find a unique state due to all the symmetries and everything you can find a unique value for the ground state at the point zero and then also I think to the first order but then there might be more possibilities in order another approach has been to deform the model so what you can do is somehow to single out some of the directions and make a deformation and then arrive at a different model which still has the same complicated spectrum a continuous spectrum but it has somehow reduced in complexity on some other aspects so this might be more amenable to computations and another approach is to somehow average with respect to symmetries so again you can view the model in a certain direction understand a certain region or selection of the coordinates as essentially also a harmonic oscillator type of problem and it turns out that if you take this operator which is then defined in this direction and you somehow averaged it over the directions you will find the full operator so the hope is to be able to use some averaging techniques to arrive at the model and then another approach has been to try to investigate this case whether the zero and new states just in the Hilbert space but maybe the sort of weakly in the Hilbert space so that if you just change your normalization a bit or if you allow for a slower decay or you know either just having constant functions or a slow decay that you might then spot these states and can somehow understand them that if you yeah that perhaps this is just a dimensional issue or that the decay is somehow there because of the geometry and then so maybe there is both in the two dimensional case and in the nine dimensional case there could be a ground state but it could turn out that it's in the higher dimensional case it's fast enough ok så maybe I just flash some things here this concerned the construction by recursive methods essentially expanding the state to higher orders in the coordinates and then considering the various conditions arising from it being a zero and new state and you can find that in the nine dimensional case with only well n matrix size is two then you can understand this essentially the value at the origin due to symmetry so it has to be a certain combination of states arising from the different representations here go into the detail of this and also Mikrochita and Chachilevsky studied the higher orders for this problem the deformation case so essentially it's a little bit related to this what happened in the three dimensional case that is somehow considered a cubic superpotential and then where we somehow use we single out some directions so let's say the last two directions eight and nine and also the indices eight and sixteen and then somehow you can using that choices those choices you can write down something which is nice and then you can somehow deform your model with respect to this choice and then you end up with so there is a deformation parameter mu and you end up with a Hamiltonian which then depends on this mu and in the case that mu is equal to one some of these terms drop out and you have something which you can which is in some sense simpler but you can still prove that this deform thing also has to the continuous spectrum then there is averaging or maybe not say so much about this what I can say you know so I have three minutes ok so maybe I can say a little bit about this weighted approach so as I mentioned the asymptotic analysis suggests to allow for more slowly decaying ground states so we saw already in the D was one this sort of degenerate model in some sense there was this ground state which is just constant so so the question is ok maybe it's just that it's not decaying fast enough so if we so if we change the Hilbert space a bit and we can accommodate for such states so here I change it by adding a weight so it's the original space so d rd times n square minus 1 with values in the form of a fox space but then I change the measure here a bit so I take this function raw alpha which is essentially just something which is decaying as x to the minus alpha so alpha is some weight parameter that can play with so in this so the new Hilbert space is just related to the old one with this function raw alpha so then we can define Hamiltonian with respect to the new space so h alpha but we define it just using the old quadratic form essentially it's still q times the q acting psi in the old norm so essentially you define this as an operator just say you start with the smooth functions with compact support and then you take this is a non-negative form so then you can take the corresponding fringe extension here but then there is an interesting ground state correspondence here so say that we had a state which is a ground so if it was a ground state of the original problem original Hilbert space and it's a zero eigenfunction of the Hamiltonian H but then since this deformed or this weighted Hilbert space it's just bigger you allow for more things here so then it will also be in that space and it will because it's zero eigenfunction here so it's annihilated by q so it's indeed also annihilated by this side so any ground state of the original problem is also a ground state of the weighted problem but on the other hand if you have a ground state of the weighted problem then you can you know that this is annihilated by q and you can use essentially elliptic regularity and these things so that you can conclude that this is actually a smooth function and it's annihilated by q so okay and it's also in that weighted space but it could be so you have essentially everything you want but it might not just be in the Hilbert space so it's a useful thing to look at this space instead to see if there's just some weakly bound states somehow in this problem so then what you can use is a certain spectral relation between the problems so if you want to see you know study the spectrum of the of the weighted problem it's the same as studying the spectrum of the original problem where you deform the Hamiltonian by say lambda times this function rho alpha so essentially if you want to see to understand whether this has a discrete spectrum then you just have to sort of look at the negative eigenvalues of this problem and so and also the question whether there is zero any ground state for the weighted problem is the same as whether this original Hamiltonian has a negative eigenvalue when you deform it in this way so you throw in a negative potential which has this decay so x to the minus alpha at infinity and you ask if there is always a negative there then it means that this weighted problem also has a ground state normalizable ground state so then you can sort of use spectral theory for these operators and so on so what I did was to look at the toy model again so this one and apply this procedure and I find that if the weight is large enough so if the decay is fast enough so alpha bigger than 2 then indeed this problem the number of negative eigenvalues is bounded by constant so this means that this in this picture that there is a discrete spectrum for this model okay so maybe I can I can sorry yeah so I wrap up maybe I can just say that in this weighted picture you can consider essentially a weighted index try to count how many more states do you get in this in this in a weighted case than in the original case so that's one possible approach so then there is just this final slide as I mentioned continue to construct the ground state to higher orders around the origin study this deformed operator there is also the question of the averaging of eigenstates the yeah so I have not proceeded yet with this approach to compute the weighted index for the toy model and the matrix models that would be interesting as well and then we saw that there are embedded eigenvalues in certain dimensions but what about the two and nine dimensional case at least in the two dimensional case I think there is some kind of reduction you can make also with respect to symmetries so that you can in some cases you can find I think you can in the two dimensional case also discover a sector of the Hilbert space where you have a discrete spectrum so thank you very much just a comment because it happened to be the end of your talk this D equals two case I don't remember so there is a paper by three former Soviet Union one of them is the mother where is Antal the mother of the woman who is faculty in Browen what's her name Arafi right I don't remember the time but a nice paper about embedded eigenvalues in D equals two this is exactly the one I was thinking of Arafi var koscheleven Medveten any questions, comments yes so this was very nice and a lot of information so what is the final message are you saying that this waiting could allow to find finally interesting ground states, weekly decaying I got lost what is the main message exactly the main message is that things depend on the dimension somehow it could just be that that even if there is no state it could still be that it's not normalised there is a state which is somehow decaying but it's just weekly decaying so then maybe you can compare the two in the nine dimensional cases so I think both in the I would somehow suspect but I don't know exactly what to ground it on but that both in the two dimensional and the nine dimensional case there is such a weekly such a weak ground state but that in the nine dimensional case it's not only a weak state but it's also normalisable with the original Hilbert space zero energy ground yeah, zero energy and that would also be embedded in still the continuous spectrum of the weak Nikolai yes, yes it's just in the end point of the spectrum one more question physics wise what's the message we have a sample of mass motion that is just inertial I guess and is it crucial to have a ground state or is it not crucial or what what's the physics message so this possibly someone else should respond to but I think yeah so I'm not sure exactly why you want to have a certain or what the motivation is to insist on a certain Hilbert space or if you have some interpretation if it turns out that your ground state is just not barely normalisable somehow that maybe you still have an interesting interpretation of it does it mean that there is for example a continuous mass spec that one I think is more in terms of this case that you can deform the membrane that you have these spikes and tubes and so on as I understand it this continuous spectrum indicates that you can easily move between different configurations but then I don't know I don't know exactly what if there's embedded eigenvalues this is somehow stable more stable but it seems it can still deform I think what you need in the end is really to just split up the Hilbert space into the different sectors and so you have some of these perhaps on the so we have the SUN symmetry but there's a constraint so everything should be there but then we have the spin D and that's not a constraint but a symmetry but there's also part of the conjecture is also what about the symmetry of the ground state so I think it is known that it should be a ground state should be spin D symmetric somehow so I think if you split the whole Hilbert space down into different sectors with respect to the symmetry then yeah somehow there's also a question whether these these spiky states these states you use to prove the continuous spectrum maybe they are not respecting the symmetry actually so maybe that's just a sector which you can remove completely last yes BFSS wanted the ground state same the ground state for some duality reasons but maybe Pilgen has the best comment you know these things although we motivated here as a regularized dynamics of membrane this also appears essentially the same thing but in the so called M-theory hypothesis there is 11 dimensional theory that reduced to 10 dimensional string theory and there is so called Kalluzer applying mode all along this compact circle so this in case of SUN N by N case this unique state you are looking for is exactly this Kalluzer applying particle so this conjecture I mean not only any 16 16 supercharge but 8 supercharge 4 supercharge come from essentially M-theory and type 2 super strength compactification so there is a large number of physics ideas that goes into this conjecture maybe we thank our speaker and I will continue