So I would like to thank all of the organizers. It's very nice to be back here in IHS and for putting together such a nice group of people.

And I should also apologize on two points. So one is that it has been a number of years since I was working actively

on this topic, so my memory is maybe not as fresh as it should have been otherwise. And the other one is that I was hoping to have a last week to get a little bit more into it, but then I had a cold, as you might recognize. So you have to bear with me.

So I plan to focus more on the spectral properties of the membrane matrix models and discuss the difference between the classical side, the quantum and the supersymmetric matrix models that arise from the membrane.

So the plan is to give a little bit more of the introduction. So we have already heard many of the talks have focused on bringing to attention the different aspects of these models, but I think it does not hurt to point out a few extra things.

And also I believe that in this audience there are several people who are not experts in this area, so I think it could be nice to understand this in more detail. And then focus around the spectrum and in particular the ground state conjecture.

So this is concerning whether the lowest point of the spectrum, which turns out to cover the positive real axis, whether the point 0 is an eigenvalue or not, whether there exists a normalizable eigenstate. And then I will discuss various approaches to the study of ground states that have been

many over the years and have been involved in some of these and provide some outlook. So if we recall, we are considering a bosonic memory. So let's start with the bosonic case. This just means that we have an embedding of a membrane in world volume into space-time.

So the picture one could keep in mind is that we have, okay, this is some surface and then we have some time dimension, which is sweeping out world volume.

And so sigma is fixed. Let's fix the topology as a start and then 2D compact manifold. And then we have the embedding coordinate functions, so X. And here I'm working in the light front, the light coordinate framework.

So even though the embedding space-time is R, so 1 plus little d will be my parameter in these models and then one time dimension. So d is essentially the transversal coordinates to the membrane.

So we have d such embedding coordinate functions into one for each of the coordinates. And then there is this issue of writing down the Hamiltonian.

So this is after having done this reduction into the light front coordinates and looking at the internal energy on the membrane. So this energy is given by this Hamiltonian function on the phase space.

So we have these X coordinate functions, but also there are the conjugate momenta P. So this is then just an integral over this fixed surface here. If you want to think of it as having, maybe there is some hole in it.

And we have the momenta squared and then there is this Poisson bracket. So you have the Poisson bracket of the coordinate functions squared. So sort of provide for you the volume of this thing.

So we have essentially two Poisson structures here. One is given by the surface sigma. Here I might have set this normalization to one, this density to one, but there is some flexibility there.

So this is one of the Poisson structures that we use, which is canonically given since it's a two-dimensional surface. But then we also have the dynamical Poisson bracket. So these coordinate functions and the momenta should be canonically conjugate.

So that's this Poisson bracket in the end here. I'll be normalizing this way. And I can say also that some of the details concerning the dimensional reduction from the full theory down to this setting. You have to eliminate some constraints and so on. There have been some discussion in a paper with De Vool and Hoppe.

Then we go over to the matrix context. And as we have seen, we should then replace this Poisson algebra of functions on the surface.

Actually they have also been normalized, so we have taken away essentially central mass and so on. So that they are zero mean real valued functions, xj. And then we represent this Poisson algebra in terms of matrices. So we're then going over to an algebra of traceless, Hermitian and Bayesian matrices.

So the tracelessness is from the zero mean and the Hermitian from the real valued functions. And then it's just n is the matrix size. And then in this representation, we represent the Poisson bracket by the commutator of the matrices. And the integral on the surface goes over to the trace.

And then the nice feature here is that there is a convergence. So we can essentially, a little bit independently of the actual topology and so on of the surface, we can arrange a basis of our traceless Hermitian matrices,

such that the structure constants in this basis converges to the given one. So if you wanted to use higher genous surfaces to parametrize your membranes, how would that show up on the right hand side?

So then I think you would need to choose a different basis which gives you those structure constants. But maybe Jens would be the most suited to... It was discussed earlier, so this answer is the following.

There is an abstract theorem, I mean, many different aspects, but one, for instance, who proved that for any genous surface, the matrix algebra can be taken to converge to the genus G surface.

But it is not known what basis to explicitly take for higher genous surfaces. I mean, these matrices in which you get the structure constants F, A, B, C are only known for tori and spheres.

So there's an existence, there's a sequence of basis, and thus a sequence of structure constants in that basis such that you get convergence. But the details are a bit more tricky. But the whole idea here is that this representation somehow respects the symmetries,

or the constraints of the theory. So we had in the original membrane theory, we had diffeomorphism invariance on the surface, and actually area preserving diffeomorphisms, because we want to conserve the area.

And then this in the matrix model side is represented as a SUN invariance, so the special unitary because of also this area preserving constraint. So we have then the Hamiltonian, which we have seen,

as now a trace over the momentum side of the matrices squared, and then you have the commutator squared over all the pairs of the matrices. And if you want to write this down in terms of a basis, say of SUN,

and some standard basis, then you have just the sum of the momenta squared, and then we have the structure constants, and then this will be the commutator squared. And then still we have 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 since we have a finite dimensional system, we can then quantize this in the standard way using the Schrodinger quantization.

So we represent this set of coordinates and momenta on the square integrable functions over the corresponding dimension. So we had the d space dimensions, and then this is the matrix dimension,

the number of elements of the 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 the derivative with respect to xj, in order to represent 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 may be represented within this. Essentially, Hamiltonian commutes with such rotations, but it also commutes with rotations in the matrix space. So it's a SUN transformations, which are then represented in this basis

by orthogonal transformations as well. So what we have to remember is that we have these constraints of diffeomorphism 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 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 the constraints. And this just amounts to the standard Dirac constraint quantization procedure that we still have some constraints. So we will look to solve those constraints on the quantum side. This is the constraint? This one?

This one is a constraint, yes. First class or second class? This is, at this point, I don't remember. Yeah, it could be a second class at this point. So yeah, you have to, this is probably written in this paper.

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 there's a Hilbert space, there's a grading operator k, there's a Hamiltonian, and then there's super charges, super charge operators. So you have a Hilbert space.

The grading operator is just an operator, a bounded operator is going to one. So you can sort of split the Hilbert space into a positive and negative part, or the eigenspaces 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 these super charge operators, qj, and if there are n, the curly n of them, 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 interchanging these spaces. And they should satisfy essentially Clifford-type algebra. So k square 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 a square. OK, I should say 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 the splitting,

so you can say that you have an even state, for instance, with positive energy, then by just acting on it with the super charge, 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.

OK, and then 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 non-trivially in our space.

So we are considering actually representations, both 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 Fock space,

or the Hilbert space of the model. So we then have to go to Clifford Alibas to construct such representations. So starting with the d-dimensional space, and we're looking at corresponding Clifford Algebra 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, the matrix degrees of freedom, and then you construct a Clifford Algebra on top of that space. So then we go up in dimension a bit more.

And then you consider the irreducible representations of that algebra, and that will be even bigger. So then the dimension is two, and then it's essentially this dimension here, Nd times N squared minus one, and then half of that. And we will see explicitly how you construct these things in one way.

Really a real representation of the gamma matrix? Yes, so here 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, the first stage is...

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 quaternion representation, and then you can use that. But on this stage, you can also ask, should you have a real or not?

But let's think of just the complex case in this 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 kinetic energy essentially plus the scalar potential, we add these spin degrees of freedom

or these fermionic operators, the theta. And they are coming linearly in the coordinates x. The rest is just these 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 linear in momenta and then a quadratic in the x's

and then coupled to these fermionic variables. And then it turns out that for certain dimensions, these satisfy this supersymmetry algebra that the anti-commutator of such supercharges gives you the Hamiltonian. But there will be an additional part here. But it turns out that on the physical Hilbert space,

so this is just the generators of sum. So on the physical Hilbert space, this piece vanishes. So it closes up to this supersymmetry algebra. So in order to understand this requirement, one needs to know a little bit more about Clifford algebras and so on. But I don't have so much time to get into it.

But I think it's useful to have sort of this table. So this is the dimension, this Rd. And then you consider the corresponding Clifford algebra. And these are well known to be essentially matrix algebras or a sum of two matrix algebras. But the matrices have a real or complex

or quaternionic structure. So this is just a dimension of the Euclidean space that we construct this Clifford algebra over. And then this is the dimension of the irreducible representation of the Clifford algebra. And you can read off from this side.

Then I have essentially just written what sort of the structure is 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 this supersymmetry is the case 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's 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's 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 algebras are the real complex quaternionic and octonionic ones and it's sort of the same thing which appears here which has to do with Clifford algebras.

So I might come back to this table. So now our full Hilbert space is then the bosonic one times the fermionic Fock space where we have represented these Clifford algebras. And then again there's a physical Hilbert space

where these constraints vanish but now it's not only that bosonic sector but there's also a 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 or spin D symmetry which is then essentially a rotation among the coordinates and then also a rotation on these Clifford algebras around the fermionic Fock space. So what I would like to emphasize here

is also how we can construct these Fock space representations. So essentially the one way which was used by is to essentially you pair up, so given these thetas, it's Clifford alga,

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, annihilation, fermionic operators satisfying canonical algebra like this.

And what has been done essentially is 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.

We can take the real and imaginary part of that. So I've here written x prime as 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. So then there will be a part which involves these first variables here and so big gamma is sort of the reduction of these gamma matrices in terms of the splitting. And then you have the last part.

So this first part sort of preserves the fermion number, but in the last two parts, you have a raising of two or lowering of two fermions. So in this sense, the whole operator will sort of mix the fermionic sectors.

But there's 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 Clodson and Halpern and then discussed this a little bit in my thesis that in these special cases,

since you have this complex or quaternary 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,

so your Hamiltonian is then the bosonic and then you have a piece which conserves the fermion number. And then the SUN and the spin D generators are 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 matrix degrees of freedom. And then there's a piece which vanishes on the physical Hilbert space.

So one way to, the corresponding supercharge, so there's only one supercharge in this case, it's essentially just a theta times the derivative. So this is like a Dirac operator and its square is this Hamiltonian on the physical Hilbert space.

But there's also here an alternative. So either you sort of work with this or you construct what you can call somehow comological version, which is, so there's a way to also introduce fermionic equation 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 super symmetry algebra. Let me not spend too much on that. So now we come to the question of what about 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 one way to understand these differences is using toy models, which has been very fruitful. So if we start with the classical model,

we have this Hamiltonian again, the 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, let's see if I, maybe I can write it 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 some 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, transversally to 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 and steeper 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 unconfined potential. So on the quantum side,

we are then considering this Schrodinger operator, essentially the Laplacian here, plus this scalar potential. And then again, both of these operators are 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 the Laplacian in the two dimensions, plus 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. 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 split your potential into two pieces,

half and a half, and then you take also half of the x momentum, you 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, is just like a harmonic oscillator. In the x variable with a frequency y here.

So this is just bounded from below by the zero point energy, which is then the frequency y. And the same thing here, it's 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 on how far out you are in the valley essentially. So bounding from below with these parts,

you get in total an operator which has a potential which goes to infinity in all directions. Because of this, it has a compact resolvent, so it has a discrete spectrum.

And also it cannot have a zero eigenvalue. You can see this, for instance, by Sobolev inequality or something like this. So this is the interesting feature here that on the quantum side we get actually a discrete spectrum for this operator. The same thing goes through, as I mentioned,

with these more complicated potentials. So essentially you can parametrize 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 and then there's this matrix here also. What we already know is due to the supersymmetry

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 for the supersymmetry that it turns out and 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 we had on the R2.

Then there's 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

with values in C2. So you have the Pauli matrices acting here. I'm sort of confused why you say minus square root of x squared. Yeah, okay, so 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 is just x squared plus y squared due to the Pauli 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 it is, okay, it can become arbitrarily negative but essentially if you go along a potential value,

so 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 a discrete spectrum

but it's rather a continuous spectrum on the positive real line for this operator. Of course, it depends on which operator we consider but this toy model has the same 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 an essential spectrum from zero to plus infinity. So what you do is essentially to use this fact that, so you put,

so on the fermionic side you put yourself 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 wild, a wild sequence of states. So you can prove that there exists a sequence of smooth and rapidly decaying states.

I think maybe you can even, okay, maybe not compactly support, well you can take them composite supported and normalised such that for any lambda greater than or equal to zero the Hamiltonian minus that acting on these states goes to zero.

So this means that this point lambda is in the spectrum. The theorem is for the full... This is for the full, exactly. So this was in De Witte, Lüschen and Nicolai where they also used this toy model to illustrate the point.

So in the toy model case as I said it's Xi should correspond to putting yourself in the lowest energy here and then this Phi is essentially the eigenstate in the transverse direction and then Xi is used 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 deform everything. You can move into this valley with zero energy cost. So in that sense that...

So what it means in this valley if I understand correctly is 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... From the fermionic 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 and it's just connected by this little tube. So somehow you're 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 the topology and everything. 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? So whether there is some stability in this sense

that you will have a lowest energy state and everything can somehow boil down too. Or if there is no normalizable ground state there it somehow tells you that things are leaking out to infinity.

There's 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's a uniqueness there. So that there's only one such state.

While in the lower dimensional models there should be no normalizable zero in this state for any matrix size. Also this statement was for all matrix sizes. By the way, you separated the center of mass motion. Yes, so this is now... So if I have a k squared called zero ground state

I would not see it. If this is a massless ground state I cannot go to the center of mass, right? Is it excluded by the formalism? I think that problem probably trivializes

the center of mass problem. You have somehow separated these things. So I think on the center of mass side this is probably just like a particle. Gauges for the string when one quantizes the string if you do it badly you eliminate the massless concept.

But then... Yes, excuse me. 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 Jurgi's relation in your software. 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 somehow the size of the... We will come to this, but there's sort of the geometry involved here

whether these states are essentially decaying fast enough at infinity. And this question was then addressed I guess it was discussed somehow in Halpern and Schwarz, but then Freilich, Groff, Hasselaer, Hoppe and Yau considered the asymptotics also in the n equals two case,

essentially studying what happens along these valleys and looking at the equation. I think that this is a zero and near state with respect to the super charge

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 normalizable or does it sort of, maybe does not even decay, it could also blow up or just stay constant

and then you have to be, you have to sort of know also the size of these valleys and so on. Interesting geometry in this. Then I guess Pelin Gee 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-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 three and five case, it 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 in 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 exists embedded eigenvalues in those models. But this was in these cases, the three and five. So the problem is in the 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. So that was what I'd like to point out here,

that on the right Fock space sector, we have the same Hamiltonian here. So another thing which is important to point out is that in the three dimensional case, 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 can actually be written as essentially a direct operator but then conjugated with a super potential essentially, which is of a cubic type.

So in the three dimensional case, you have also more canonical pairing up between all these, essentially form a 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. So 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 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 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 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 or the super charge is just essentially a derivative

times a fermionic variable. So again, you can just take states which are constant with respect to that 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, it's a constant so it's not in the Hilbert space but it's better than this case because then it's constant so if you just change your inner product a little bit you can somehow accommodate for it. So this is an interesting question whether one should somehow change 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 approach is by construction.

So you write down recursive equations on the different sort of expanding. So one approach is to expand, essentially tailor 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 if you go up in order. Another approach has been to deform the model. So what you can do is somehow

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. You can view the model in a certain direction or understand a certain region or a selection of the coordinates

as essentially also a harmonic oscillator type of problem and then 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 or any states just are not in the Hilbert space but maybe they're 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 either just having constant functions or a slow decay that you might then spot these states and can somehow understand them from that perspective.

Perhaps this is just a dimensional issue or that the decay is somehow there because of the geometry and then maybe 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 that the decay is fast enough. So maybe I'll just flash some things here.

This concerns 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 energy state.

And you can find that in the nine-dimensional case with only n matrix sizes 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. We'll not go into the details of this. And also Mikhalshita and Czachelevskii 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

say the last two directions eight and nine and also the indices eight and 16 in the Clifford Algebra and then somehow using 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's a deformation parameter mu and you end up with a Hamiltonian which then depends on this mu in the case that mu is equal to one some of these terms drop out and you have something which is in some sense simpler

but you can still prove that this deformed thing also has the continuous spectrum. Then there is the averaging maybe not say so much about this and what I can say

so I have three minutes 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 equals one

this sort of degenerate model in some sense there was this ground state which is just constant so the question is maybe it's just that it's not decaying fast enough 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 or d times n squared minus one with values in the formula in the Fock space but then I change the measure here a bit so I take this function rho alpha

which is essentially just something which is decaying as x to the minus alpha so alpha is some weight parameter that you can play with so the new Hilbert space is just related to the old one with this function rho alpha

so then we can define a 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 acting on psi in the old norm

so essentially you define this as an operator just say you start with smooth functions with compact support and then you take this as a non-negative form so then you can take this corresponding free extension here

But then there is an interesting sort of 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, so it's in the 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 a zero eigenfunction here, so it's annihilated by Q, so it's indeed also annihilated on 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 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. And it's also in that weighted space. 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 study the spectrum of the weighted problem,

it's the same as studying the spectrum of the original problem where you deform the Hamiltonian by lambda times this function rho alpha. So essentially if you want to understand whether this has a discrete spectrum, then you just have to look at the negative eigenvalues of this problem.

And also the question whether there is a zero energy 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 eigenvalue there, then it means that this weighted problem also has a 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 two,

then indeed this problem, the number of negative eigenvalues is bounded by a constant. And so this means that in this picture that there is a discrete spectrum for this model. Okay, so maybe I can, sorry?

So I wrap up, maybe I can just say that in this weighted picture you can consider essentially a weighted index to sort of try to count how many more states do you get in a weighted case than in the original case. So that's one possible approach.

So then there's just this final slide. As I mentioned, one can continue to construct the ground state to higher orders around the origin. Study this deformed operator.

There's also the question of averaging of eigenstates. 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 sort of 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's Antal? The mother of the woman who is faculty in Brawen. What's her name?

So there is, I don't remember the time, but a nice paper about embedded eigenvalues in D equals two. Yeah, this is exactly the one I was thinking of. Ah, yeah right, you remember. 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 weighting could allow to find finally interesting ground states weakly decaying, because I got lost at the end. What is the main message?

So exactly, the main message is that things depend on the dimension somehow and it could just be that even if there is no state, it could still be just that it's not normalized. There is a state which is somehow decaying, but it's weakly decaying.

So then maybe you can compare the two and 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 both in the two dimensional and the nine dimensional case there is such a weak ground state,

but in the nine dimensional case it's not only a weak state, but it's also normalizable with respect to original Hilbert space. So the zero energy ground? Yeah, a zero energy state.

And that would also be embedded in still the continuous spectrum of the weak Nicolai? Yes, yes, it's just in the end point of the spectrum. One more question. So physics wise, what's the message? We have a center of mass motion that is just inertial I guess,

and is it crucial to have a current state or is it not crucial or 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, it's just barely normalizable somehow,

that maybe you still have an interesting interpretation of it. Does it mean that there is for example a continuous mass spectrum? That one I think is more interpreted 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 exactly 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. So you have some of these, perhaps on the, so we have the SUN symmetry, which is actually not just a symmetry, but just 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 you would, there's also a question whether these spiky states,

so whether 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. The EFSS wanted the ground state and claimed the ground state for some duality reasons,

but maybe Pilgin has the best comment, you know these things. Although we motivated here as a dynamic, regularized dynamics of membrane, this also appears, essentially the same thing, but in the so-called M-theory hypothesis, there is an 11-dimensional theory that is reduced to 10-dimensional string theory,

and there is so-called Kaluza-Klein mode, momentum mode, all along this contact circle. So this M, in case of SUN, N by N case, this unique state you are looking for, is exactly this Kaluza-Klein particle.

So this conjecture, I mean not only N equals 16, 16 supercharged, but 8 supercharged, 4 supercharged, come from essentially M-theory and type 2 superstring compactification. So there is a large number of physics ideas that goes into this conjecture.

Maybe we thank our speaker and then continue.