 Ok, so it's a great pleasure to give this talk in honor of Alan Vallette, who as Claire just mentioned is a very brilliant student that I had at some point. And I mean it's quite striking to notice the conjunction between Alan and Yeshua. Because they are both 60 this year. So which means that they must have been born more or less at the same time. In the same room. Yes, and also I mean it's quite striking that Alan has played a very important role in the administrative part of the Council of Yeshua. And it's also quite striking to see that his mathematical inspiration is very very close to few of the great people in Yeshua like Misha and so on. So I mean he has both this taste for geometry and analysis and so on that we all appreciate very much. So I will come back to that at the end. Ok, so today what I want to explain is quite simple. It's a joint paper with the Shamsedin and Sahelokalman. And if you want it's a strange thing because it is the fact that the spectral action which I will explain is in fact surprisingly related to entropy. But through this relation so I can tell you right away what is the point and if you want what is the main result. So the main fact is that through this link between the spectral action and the entropy. Normally when one defines a spectral action one uses an arbitrary test function. So I mean the spectral action is something which is defined for spectral triples. I will explain what it is by some function of the operator D which defines the metric. And a priori the function F is like an arbitrary test function. But in the link with entropy which I will explain the function takes very specific form. And what we were quite amazed by when we wrote this paper was that this function is in fact intimately related to the Riemann Zeta function. So this came out as a total surprise. And so if you want the motivation of all this work which I have been doing with Shamsedin, Mukhanov and Sahelokalman and several people, the motivation came from the idea that it's probably a misconception to think that when we look at space or spacetime it is the continuum at any scale. And the person that actually already said that from the very beginning is Riemann himself. What he said when you look at his inaugural lecture on geometry, what he said is that if you take the notion of a solid body or of a ray of light on which he was basing if you want the Riemann paradigm of ds squared equals g mu dx mu dx nu. So what he is saying is that in general you cannot rely on this notion of solid body or ray of light in the infinitely small. And in fact you have to be careful. And so what he says is that it's therefore quite conceivable that the metric relations in the very small do not conform to what he had set up as the hypothesis of geometry. And then he goes further actually and he explains that in fact if this is a case when you look for the foundation as the forces, the binding forces which if you want hold the physics and they should be involved in the geometry of the basic geometry if you want the geometry of space or spacetime. So in fact it turns out that quantum mechanics gives forces you to take a paradigm for geometry which is more flexible than having continuum and having the g mu dx mu dx nu locally and so on. And this paradigm took some time to reach and its notion of spectral geometry And spectral geometry is given by a triple. So you have three ingredients. Well in fact you have two ingredients which are in the middle which are acting in the same space. This is the Hilbert space, infinite dimensional countable basis. This is an algebra. So it's an algebra but which is acting as operators in this Hilbert space. And this guy, this d is a symbol for the Dirac operator. This is what you have to keep in mind. But it will be if you want the inverse line element very soon. So it turns out that this notion of spectral geometry it has three aspects and it would be a mistake to take one of these aspects as being the main one because each of them plays a role. So the first aspect is the k-homology. And this aspect in k-homology is quite rooted in mathematics. And this means that when you consider if you want the data which is here it has meaning because when you have a manifold it's not just space up to a motopy. It does characteristic classes like the Pontragon classes which are finer than just giving the space up to a motopy. And this wouldn't be seen if you take ordinary homology but it is seen when you take k-homology. And essentially this triple, if you want that part the k-homological part gives you the fundamental class. Now the second aspect is the metric aspect. And it turns out that in metric aspect instead of measuring distances by taking an arc between two points and taking the smallest length of the arc using the square root of the square you do something different. And the fact that you do something different allows you to measure distances in spaces which are for instance totally disconnected which don't have any arc. But the distance gives you the same formula when you are dealing with an ordinary space and the duality between the two formulas is the contour of each type. Duality which is familiar in the transport equation a transport of stuff like a mange and so on. So this is the metric aspect. The metric aspect is quite important. Quite important because it will make sense in a commutative case as well. And finally probably the topic that will be most important for today is the link with physics. So in the link with physics what happens is that physicists in the quantum they have what are called Feynman diagrams and I mean when you consider Feynman diagrams which have to do with fermions then there are things which relate two very close by points together and this is the fermion propagator and it turns out that in a commutative geometry I mean with the spectral triple sorry this is really the line element. So it turns out that it fits it fits with a lot of things and we shall use this link with physics when we find the link with entropy. Okay so I'll go very fast in the first part because this is not my point today I want to explain the link with entropy so I will go very fast just to tell you what is the framework, the general framework just to survey the general framework. So as I said you know the notion of manifold is more delicate than just having poincare duality in ordinary homology and I mean this comes essentially from one result which is very striking which is due to Sullivan and which says that if you take a micro bundle so the micro bundle will exist as soon as you have a space which has poincare duality in ordinary homology if it's simply connected you put it in a usual RN space it has a normal bundle but it's not a bundle it's a micro bundle and if you could make this micro bundle into a bundle, vector bundle then you could use term transversality to make it into a manifold but in general you cannot and what is missing is exactly what is needed for a micro bundle to become a PL bundle let's say which in dimension 4 is the same thing which is what is called the k-orientation so this is the usual stuff now this stuff is purely topological and that stuff which was purely topological becomes now related to in the space because of the work of Attia and Singer on index theory so both Attia and Singer understood that there was an abstract way to formulate index theory which was with abstract elliptic operators which are acting in the in the space and then gradually through the work of many people Brown Douglas Fillmore and finally Kasparov with the Kasparov theory it turned out that this allowed to reformulate the dual theory to k-theory which is k-o-homology in terms of Hilbert space so this accounts if you want for the homological part and so I mean one has a very simple dictionary so in that dictionary in the classical column what you have is the usual Riemannian formulation and what you have in the other column in the quantum column you have the translation in the computer geometry but as I said I mean the main point is that now you have abstracted the idea of manifold and you have also abstracted also now the possibility to use an algebra which is no longer commutative that makes a big big difference that gives you more freedom and now what turns out to be the case is that the square of the line element now we have been able to extract the square root this extraction of square root is the extraction of the square root of the Laplacian which replaces the Laplacian by the Dirac operator so that was fine the extraction of the square root but then if you want when you think about the general framework and you try to apply it to physics and so on you wonder how can you understand if you want the role of the algebra A and it turns out that the way this role was unveiled if you want was by doing exactly the same thing for the position variables for the coordinates if you want as the Dirac operator does for the momentum so the Dirac operator it assembles the momentum into a single thing which is the Dirac operator and in the same way what we did with Shamselian Mukanov was to assemble the position variable exactly in the same way by a Feynman slash so the Feynman slash means that you add you put gamma matrices the idea is very simple for instance instead of thinking of four coordinates you can think of a single quaternion that's very nice because then you can manipulate with four coordinates as a single symbol as a single element so the gamma matrices are exactly playing that role and then it turned out that what we did was to understand a geometry as an irreducible representation and an irreducible representation of what the simplest example of spectral geometry is obtained by the circle of lengths 2 pi how do you get the circle of lengths 2 pi out of an irreducible representation well you consider the following thing you consider a unitary operator you look for the matrix so the matrix is this D you don't know what it is but you write a presentation you write an equation and the equation that you write is you start the commutator U is equal to 1 this is a kind of Eisenberg commutation relation you spell out the irreducible representations it's not difficult an exercise because why is it an exercise because this is telling you that the operator of these two guys this is a self-adjoint operator this is unitary now this self-adjoint operator will have a spectrum but this spectrum has to be if it's irreducible it has to be the integers up to a shift why because D minus U D U star I'm messing it up so it's U star D U minus D is equal to 1 so this U star D U is equal to D plus 1 so this is the spectrum has to be shift invariant by 1 so then you get these irreducible representations and once you have these irreducible representations you can compute the distance the matrix and everything and you get the ordinary circle but with lengths 2 pi now this is very simple so this is what is written here this is very simple but what is much more interesting now well here you see in this example the algebra was commutative now what is quite surprising and beautiful is that you can get much more interesting algebras than the circle I mean much more interesting spaces than the circle but by doing the following so now what you do is you take 2 by 2 matrices so you take M2 of C and to this algebra now I add a punctuation symbol by punctuation symbol I mean a Y which satisfies Y squared equals 1 and Y equals Y star so it's self-adjoint and now you compute so these things don't commute so the Y doesn't commute with M2 of C they don't commute, so you form words individually taken they are nothing 2 by 2 matrices is very discrete space finite dimension also same for Y, it's nothing but when you put them together and you write words and so on what you find is 2 by 2 matrices of functions and no relation among these the only relation is this relation namely that the trace is 0, that's all so when you compute with that you find 2 by 2 matrices of functions versus two-sphere so what I am saying now is that it's simpler to define the algebra of 2 by 2 matrices of functions than the algebra of functions of the two-sphere what do you gain you gain something which is quite amazing what you gain is that the non-commutativity of the algebra now will give you the gauge fields automatically because when you will compute what are the matrix and what are called the inner fluctuations of the matrix, you see the fact that the algebra is non-commutative gives you inner automorphisms these inner automorphisms when you compute what they are in the case of matrices of functions you find that they are exactly maps, smooth maps to the group SUN and when you divide by inner automorphisms you find the ordinary different morphisms and moreover when you do the same thing but for matrix what you find is that the matrix now they have the gm news but they also have new ingredients and these new ingredients are exactly the gauge fields of gauge theory here for SUN so in fact what we asked with my collaborators what we ask is we ask a purely geometric question purely geometric question was the following is it possible to describe now four manifolds not the two sphere by such an equation by such very simple presentation so we started we started and we started from the analog of the relation which I had written of the relation which I had written there for the circle which was a kind of Eisenberg relation you see this is very close to the Eisenberg equation pq and so on and so we wrote an analog of this equation now in dimension n and we began to classify the solutions of this type of equation and at first we were quite disappointed because when we looked at this equation and we classified the solutions what we found is that it's only the sphere or it's disjoint union of spheres of unit volume when we talk about physics this size of the sphere will be the Planckian size so at first we were disappointed and we had found this type of bubbles I mean many little very tiny little spheres but then we realized that we had not here at any dimension n here it's very general but then we thought much more and we found out that we should have involved a key ingredient of the story because it's a nuance between k and ko theory and it's very important for instance in the theorem of Suleyman so then we wrote down if you want the ko refinement of this equation and what we found out then is that for the ko equation there was an infinitely better theorem so there is this higher equation which you don't need to know in detail but what we found out for this ko equation to have solutions is that we needed two cliford algebras and in dimension 4 we need two cliford algebras which corresponded to because it's a sphere so to 5 pluses and also the cliford algebra which corresponded to 5 minuses we needed both so then we looked at the table of cliford algebras and what we found out is that by looking at this table we got exactly the algebras that we had to put by hand to get the standard model coupled to gravity in dimension 4 so the conclusion which I drew out of that is that purely geometric model if you want to present the algebra of functions in the simplest possible manner purely combinatorially by using these symbols and these punctuation symbols it gives you exactly the algebras that we had found from a bottom up if you want story to get the standard model with the x all these mechanisms v minus c and all this stuff ok so this we concluded from that that what ok ok I was a constant center you see what you find now you have to use an action functional to get the standard model this is spectral action that I am going to discuss now ok so what was the action the dream was that the action in physics should only depend on the spectrum of the line element so not on the space but just on the spectrum of the line element ok and this was confirmed because you know normally when you write many gym news all sorts of notations and so on but it turns out that what we had found is that when you expand this function in terms of heat equation expansion and so on and so forth you find several terms and among these terms I mean there is a first cosmological term which is like lambda 4 times the volume then there is a term lambda square times the Einstein Hilbert action ok plus the x square term and then plus when you look at the coefficient of lambda 0 you find the Young Mills terms the x quartic self interaction you find exactly what you need to find ok but we were very bothered for a long time by this term lambda 4 times the volume which is like a cosmological term because it's very huge however what the equation what the Eisenberg equation does is to quantize the volume so in other words exactly as in the case of the circle which I was describing here where the length of the circle was 2 pi it turns out that when you write this Eisenberg equation the theorem is the following the theorem is that in dimension 4 dimension 4 is critical because in order to get a solution you need to cover the sphere in two ways by two maps in such a way that when you pull back the volume form of the sphere and you add the two pull back you get something which never vanishes so for that dimension 4 is critical and the reason why dimension 4 is critical is that in general you can get a ramified cover of the sphere which is ramified in a variety of dimension n minus 2 when you are in dimension n but if you take two of them then they will intersect you know they will intersect when this is n or less or equal to n and this gives you n less than 4 exactly and for n equals 4 it's very tricky actually to be able to do that it's not true for p2 of c but it's true amazingly for n is pin manifold so for n is pin 4 manifold compact what you can do you can cover so you have this picture if you want what you can do is you can cover the sphere by two maps and then because the cover can have multiplicity which is arbitrary large but it has to be at least 5 or something like that then if you want what it means is that you can get a space whose volume now with size if you want will be very large in Planck units if you couldn't get all these integers you couldn't get something out of the sphere you would be stuck with a very small sphere so now I come to the subject of my talk which is really the spectral action so spectral action and I said you know it had a bothering cosmological leading term but now this cosmological leading term is totally unimportant because the volume is quantized so this is an integer so the fact that the volume is quantized tells you that when you take the variational equations with respect to this action you won't have this cosmological term anymore ok it disappears integer means positive integer positive integer positive ok so the volume is positive integer so we were not worried anymore about this and then if you want the idea the general idea which will require a lot of work of course is that one should see the geometry as emerging from the quantum in other words all of the stuff which I have explained it's all operators in Hilbert space and so on I didn't put the geometry by hand from the start you know I'm just looking at operator equations out of the spectrum and so on emerges the geometry so this is the idea ok but now we come to this you know usually when you say that to physicists they say ok fine fine fine but what about the second quantization because what I have told you so far is really at the first quantized level if you want it's a story which is not taking quantum field theory and so on into account ok it's a story of course you know it's very important that you have a good starting point so I think we have a good starting point in order to recover geometry from the quantum but ok we need to understand how if you want how the second quantization will enter ok and so as I said you know the fundamental problem is to understand at this point of the story ok which was several years ago you understand the role of second quantization now this is the reason why I explain this stuff about case theory and chaos theory at the beginning because you see what has happened is that thanks to the index theorem the dual of case theory case theory was known before but the dual of topological case theory did appear to bring with itself you know with the suit cases the quantum because the cycles were defined by representations universe space and so on now it turns out that there is a magnificent generalization of topological case theory which is a Quillen algebraic case theory and which is a theory which is quite quite more elaborate if you want and more difficult to define than case theory than ordinary case theory and so far nobody has developed what is the dual of the Quillen algebraic case theory in fact in my inaugural talk in the college de France in 1985 I mentioned this as a fundamental question ok and so we have with my collaborators exactly you see, k homology is dual to case theory and what I believe is that the second quantization quantum theory will be dual to Quillen case theory ok and I will give you some some confidence for that ok I had at the time well ok I mean we are working overseas you see I mean ok you could worry about non-archimedean places but we shall not do that today ok today we are over the complex numbers ok so so the reason why this is very interesting the following is that you see the one characteristic property of the spectral action of this function is the fact that if you take two spaces which are disjoint you take the disjoint sum of two spaces then this functional will be additive whatever function you take and the reason is that the operator d will be the sum like d1 direct sum d2 so the function will be you know it will be like adding the trace in two different Hilbert spaces so it will be obviously additive at the conceptual level is the key property now there is another way to find an additive function of spectral triple which begins to look like second gradization and what is it so it's you take a spectral triple ok you take a spectral geometry ahd ok and what do you do well you think of the Hilbert space now as a real Hilbert space so you take the underlying real Hilbert space to the Hilbert space H so that's what I denote by H sub r ok so this is a Euclidean space if you want and to that space you can assign a Clifford Algebra which if you want now is much more congenial for Neumann Algebras and so on but I look at it as a c star infinite of course so how you you know you see you take the well you take topology you take the norm which is uniquely defined by taking all unitary representations of the Clifford relations ok so you see you have vectors you have gamma of psi for every vector here and you have the relations of the gamma of psi square is equal to psi square you take representations of that ok and then if you take linear combination of these guys or product and so on you get a norm because it's a c star algebra ok so c star algebra which is well known this c star algebra is in fact a co-limit of finite dimensional c star algebra because you can take finite dimensional subspaces to be the co-limit of these and then it turns out if you want that the right way to think about it is the following is that now we have the spectral triple the spectral triple was spectral geometry but now we think of it with replace the Hilbert space by the Clifford algebra ok so we replace the Hilbert space by the Clifford algebra in the middle we replace the operator d now by a one parameter group of automorphisms of the Clifford algebra how is it done well you see I mean the one parameter group sigma t is a group which is corresponding by factoriality of the construction to exponential i t d exponential i t d these are orthogonal transformations sigma t is coming from h r automorphism of h r these are automorphisms of h r sigma t is automorphisms of Cliff ok so sigma t are automorphisms of this Clifford algebra which come from which come from the guy by factoriality ok so what I am saying is that this defines a group of orthogonal transformation which is underlying Hilbert space so I can view it as automorphisms of the Clifford algebra it came from the d in the spectral triple ok ok it comes from the d ok now you can say where is the algebra playing a role well the algebra does play a role sorry the algebra does play a role I mean the algebra plays a role because you can the way the algebra was playing a role in the previous case is called the inner fluctuations of the operator d so it's deformations of the operator d which come from the gauge potentials so this will deform this one parameter group in the same way so we have a way to second quantize our data our original data now we are going to use it and we are going to use it because of the entropy which is the fundamental entropy and the fundamental entropy which I will introduce very shortly as a property that if you take a tensor product of two situations type one the entropy is additive so the entropy of a tensor product is now the sum of the entropy so this will recuperate the additivity of the spectral action and it will tell us that when you compute the entropy of the corresponding second quantized stuff it will give us a spectral action ok so now I am coming so entropy of what so ok so we use this thing ok we have this one parameter group of automorphisms of automorphisms of the clifford algebra and so we have if you want the clifford algebra and the one parameter group that's what we have ok we have this stuff how do we get a state out of this well by the KMS condition so the KMS condition is a condition which relates one parameter group of automorphisms of the clifford algebra with a state and this for every value of beta where beta is an inverse temperature this state might exist might not exist might be unique, might be many they form a convex set I mean and so on and so forth ok there is a whole story about it in this situation where we are so this is the KMS condition if you don't know it, forget it I mean it's a condition if you want which relates normally the state which is given by trace of e to the minus beta h with the Eisenberg evolution which is given by ith dot minus ith so it's a condition that relates the time evolution with the state but of course this is a very simple case in general as I said on the C star algebra endowed with a group of automorphisms it might or might not exist so here it turns out that in our situation it's quite nice it exists and it's unique so if we take a complexive space a self-adjoint operator with compact resolvent we take the clifford algebra the complexified clifford algebra of the real number space and we take the corresponding automorphisms then for any beta positive there exists a unique KMS beta state this is not difficult to show, why because as I said the clifford algebra is a co-limit of finite dimensional stuff now the operator has this spectrum so you can take the co-limit according to the subspaces on which the operator is diagonal now you look in each of these spaces it's a matrix algebra and ok and you have a uniqueness of the KMS state so this is not difficult this is not difficult at all ok ok so you could have type 2 1 factor you could have factors of various types however in our situation where the operator has a property where the Dirac operator is finite dimensional so it has a property that if you want the resolvent or 1 plus d2 to the power minus n over 2 will be a trace class so this will be finite for a finite n so because of that because of that what happens in our case is that this operator turns out to be trace class and then it turns out that the states that you obtain on the Clifford algebra is type 1 ok I mean there are situations where it would be type 2 but here it's type 1 so the fact that it's type 1 allows us to use the fundamental entropy so what is the fundamental entropy if you take a type 1 state you can always write the state in terms of a density matrix ok so this is the fundamental sorry what is the fundamental entropy so fundamental entropy is the following you take your state you write it from a density matrix so you write your state psi of an observable a is given by trace of rho times a where rho is called the density matrix and normalized and what not and it turns out then the fundamental entropy is given by the fundamental entropy of the state so it's an information theoretic entropy so the fundamental entropy is given by minus trace of rho log ok so this is definition fine so this is where we stand now and so I mean so this is just the construction by what is called the Dirac C of the corresponding state this is standard stuff and now here is the entropy so here is the definition of the fundamental entropy ok and now we apply this to our situation we have a unique KMS state it turns out to be type 1 so we can define its entropy and we know because of the fact that the entropy for a tensor product is the sum of the entropy that it has to be a spectral action but for which function this is the issue which function so we compute and what do we find we find that because of this additivity ok it turns out that the function that we shall get I mean you know first of all let me use this notation epsilon of x this epsilon of x is the entropy of the partition for which the ratio of the sizes of the two intervals is equal to x so it's the entropy of the easy thing where you have only two two subsets ok and you assume that the ratio between the two lengths is x so I'll give you this stupid function ok and now what you do is there is the following theorem so the following theorem is that now if you do the fundamental entropy you find that it's equal to the spectral action of beta d so we have the spectral action and this is the spectral function it's h of x equals the previous function evaluated on e to the minus x ok now this is very simple it looks very very simple and what was our surprise was to find the link between this and the Riemann zeta function ok so this is really the mathematical content of the paper ok and so what did we do to do that well this is the function first of all it's a very good function you know because in the spectral action you want the function to be positive you want the function to decay very fast at infinity and you want it to be a kind of cutoff function but it shouldn't be a sharp cutoff because this would be bad you want it to be smooth yes this is the epsilon of e to the minus x first of all why is it an even function well because if I change x to minus x e to the minus x change to the inverse but I told you that this was the entropy when the ratio of the two intervals so of course you know if you have two intervals and they have a ratio of 10 if I exchange them they will have a ratio of 1 over 10 so the function is even ok so that function is even there is no problem when you look at it a little more closely this function what do you find you find that its derivative is actually without a log and its derivative is given by this formula is given by minus x over 4 the hyperbolic cosine square of x over 2 and this is where things begin to be interesting because that function you know the function being function h being even has to be a function of x square so you have to be able to rewrite things in terms of x square so this is what you begin to do you write this formula now its calculus so what you do is you write this formula this is not a big deal you check this formula with the sum over z begins to look like one dimensional kind of Heisenstein series and then what you find out is that you see you would like to write your function as Laplace transform because then you can use the heat expansion to expand the various terms so you begin to do that but not for the function itself piece of the function which was given by the square root and what do you find you find that this function g of t because of this equality is of this form so it looks like a theta function in fact it is a theta function very closely in fact so this function g of t turns out to be given by this formula and this is extremely useful because you can use the you know you can use the modularity and to prove properties so you use properties and what do you get at the end of the day you get that you can write the original function the test function which is forced on you by the entropy you can write it as an integral from 0 to infinity of e to the minus t x square very important to have x square because then we will put this square and we shall use the heat expansion of a certain test I mean of certain integration function g tilde and this function g tilde is trivially obtained from the previous is g of t over 2t ok so this allows you to go now why we can go we can go because precisely in the heat expansion we can expand the trace of exponential minus t delta delta will be the square of d as a sum and it is exactly in this coefficients that are appearing the quantized volume the Einstein Hilbert action with the x quadratic term the Yang-Mills terms and so on so we know that these terms are here we know that these terms are around but now we want to know the coefficients so we want to know if you want these integrals here we want to know these integrals and playing a little bit with these integrals what do you find you find that you write the usual formula for a power in terms of the gamma function ok and then you find out that actually you can rewrite the coefficients in the following form ok so this is just triviality ok but then we have to compute it for our specific function and when you compute it for our specific function you get the following surprise you get that the coefficient which will appear for instance in front of the Einstein Hilbert action is zeta of 3 involves zeta of 3 and the coefficient which will appear in front of the volume is this one with zeta of 5 and more generally when you compute moments ok you begin to find this strange expression where this is the Riemann zeta function ok ok so then you begin to worry because you see it should be true that this integral should define in fact an entire function of alpha so you begin to worry you begin to worry about poles about this about that ok and so I mean in the heat expansion actually this is the closed formula that you have ok and so begin to worry about these terms sorry heat equation takes place because ok in terms of the operator d you see as I said the operator d right sure it's from the spectral geometry and the operator d has in it not only the metric gminus but all the gauge potentials and so on it will spit out exactly what you want which was very very surprising at the very beginning ok so but now we look at the coefficients ok we look at the coefficients and this is the type of coefficients that we get and we begin to worry about these coefficients you know because as I said it should be an entire function so begin to worry about the poles and what not ok and you find out that there is no problem it's really an entire function so you work with it for some time you are worried you know I spent afternoon worrying you don't know whether something might be wrong in the whole thing ok and for instance you have to have rationality when you look at the expansion because the function h has only rational coefficients when you write down the expansion into the Taylor expansion except for the first term and what you find is that actually this is ok because in zeta the values of zeta at odd integers at odd negative integers are ok for that so that's what you get you get reassured but finally what you find is that the xy function of a Riemann multiplier appears so the normalization when you work in the coefficient of the heat expansion a alpha t alpha will be the following after many calculations what we find is that it's given by this elementary coefficient totally elementary times the xy function of Riemann at 2 alpha so typically if you are in dimension 4 you will have a term like a-2 t-2 as the first term then you will have a term a-1 t-1 and so on and so forth ok but with some coefficients which are these gammas and when you look at the table when you look at the table which gives these gammas this is what you find so this is for dimension 4 this would be in dimension 3 this is the term corresponding to the Einstein term in dimension 4 and then you have the other term now the very beautiful thing is the following I will finish there the very beautiful thing is that the functional equation of the xy function gives you a duality between the even dimensions and the even dimensions at large energy and the odd dimensions at low energy and conversely that's what it gives so this would be the conclusion ok so I mean so we were extremely surprised if you want we were extremely surprised because somehow before what was going on with this whole setup was that there was a freedom and there was no sort of how to say complete conceptual understanding of why we should take the spectral action while now just by touching by touching from in a very mild manner with a second quantized theory you find that there is a beautiful interpretation of the spectral action but for a very specific function and of course now the name of the game will be to see precisely with respect to the constants of the standard model how this will fit but there is if you want a freedom in this story the freedom in this story is a spectral triple that you get for the finite space so if you want the general idea what should be said as follows that what this theory uncovers it uncovers that it's probably wrong and Riemann was right to believe that when you go in the infinitely small the the structure is constantly the continuum no I mean the structure is more subtle it has some fine structure and the beauty is that this whole fine structure is encoded by the line element exactly as Riemann was saying because he was saying that the geometry should be specified by the forces which actually hold the space together that's exactly what is going on in this story then I'm not really a physicist so the issue is whether this understanding can be continued can be continued really at the second quantized level and what we are doing at the moment with Shamsedin and Sylocum we have realized that the Y which was the Feynman slash of the coordinates is actually a sigma model with values in a sphere and these are rather well understood in physics so this will give the quantization of this other side also which is not present at the moment which is totally the type of second quantization I was discussing is very simple ok so I think I will finish here ah let me now finish with the most important ok yes and you will see I am in it, it's a great time for the sixties you will see the generalized Heisenberg does it specify to the ordinary Heisenberg input no no no the pq is very very simple the pq would not allow you an arbitrary geometry I mean the beauty if you want of the equation which I wrote down is that the irreducibles are related to geometries and if you want one difficulty with this quantum gravity whatever is that the geometry has to appear from some irreducible representation here it appears from irreducible representation of this relation it's very strange it's very strange what do you mean by the second quantization the second quantization I mean the following ok mathematically speaking I mean the following we replace the spectral triples or the spectral geometry the inverse space is replaced by the Clifford algebra the real one the operator Dirac operator is replaced by one parameter group of automorphisms and the algebra itself is manifest by the inner fluctuations ok now what would be great is if a mathematician or physicist and so on would have the idea to develop this as not kaomology but you know algebraic I don't know the name that would be dual to algebraic k theory and I know this is worthwhile because I knew for a long time you know that what are called the Schringer terms in physics are dual to algebraic k2 so there is already some little int but it's another story to develop the whole thing I mean this would mean you know that physics is teaching us what is really the geometry it's much more subtle than we think of course we are just starting so physics from the quantum is sort of trying to teach us what is the geometry and it's very involved because so far if you want the picture which I told you and so on it's a one particle picture it's not the full story which is much more involved for sure no question