 It's going to be a bit different from, and I apologize also to Thibault, from most of what has been said about black holes, because there will be no, not really black holes yet. And I think the black holes and what we heard here is extremely important because black holes are things. And so in order to quantize gravity to understand this phenomenon, they will be very important. I think we couldn't have gotten into quantum mechanics without things like the atomic, the atom of hydrogen, and perhaps the black hole is a bit like the atom of hydrogen for quantizing gravity. So it's by looking at the physics that we can progress. And that's not what I'm going to do in my talk. I am a mathematical physicist, so I am thinking about the relationship between math and physics. And so what I will show you is more like an approach based on the unreasonable hope that looking and generalizing the mathematical structure behind gravity and quantum mechanics is what will lead also to progress. So in this way, I will propose that it's not physical, it's just perhaps prejudices that this physics of quantizing gravity should have a mathematical counterpart, which would be the study of random geometries. Why? Because I will take the Feynman point of view on quantization. That is quantization for me would be like integral over histories. And then these histories should naturally be associated to some kind of random geometry because you would like to make random both the metric on a space and perhaps the space itself. And the guy that we get from classical physics is that in Feynman quantization, we should use the classical action to ponder the sum of our history. So the input from classical general relativity would be to ponder this sum with the Einstein-Hilbert action or at least some approximation of. But this is very difficult. You don't know how to sum over metrics. You should be a non-renormalizable sum. You don't know whether we should include some of our space-time topologies, but I think we should first because of string theory examples where you have the sum of our genus in the perturbation theory of strings. Also because typically if you don't sum over everything that can happen, then the theory will miss a unitaryity of some key properties. But of course there is plenty of debate here. And finally, there is a huge gauge invariance which is that coordinates are man-made, but physics doesn't depend on coordinates. That's how general relativity was invented. So there is a huge gauge invariance there. And therefore either we have to fix the gauge but work in a particular coordinate system or we can perhaps take another hood which is to discretize the problem. And indeed discretizing the problem, we know it from gauge theory, is going to avoid the necessity of gauge fixing by making the also discretizing the problem if you are in front of an integral which seems to diverge in all possible ways. We've also discrete sums included there. Perhaps discretizing the problem is the best way to try to understand, to replace the problem by understanding a limit. So I would put a little bit the talk under the patronage of these two great figures of Regé and Wilson because they were both discretizing problems in different areas. Regé was I think one of the first to propose to discretize general relativity and certainly Wilson is well-known for having worked on discretization of gauge theory. But in fact the paper of Regé was earlier and he had the idea that if you look at the space with curvature and so you could concentrate, you could make it flat into simplices and then concentrate the curvature at the joints or hinges of the discretization. So then he wrote I think around 1962 the so-called Regé action which tells you this is a term which is proportional to the cosmological constant and to volumes of the simplices of your decomposition of the space into simplices and here you have the curvature term which leaves on D minus 2 simplices and in the Regé calculus the simplices that you join and which creates space, space-time, they have lengths and if you know all the lengths of course you can compute the deficit angles and then you would identify curvature with a deficit angle that is a function of the length. And I think this is a bit the same idea in a certain sense than Wilson-Lattice gauge action but it is very complicated. So the advantage is that you don't need gauge fixing and a priori if your triangulations are general it includes not only knowing the metric but also knowing the topology so the possibility of summing over topologies but it is complicated because of the length parameters essentially you have the triangular inequalities constraint and the deficit angles to compute so this is quite difficult. So maybe there is a bold idea perhaps it's not discretized enough. So this bold idea is to progressively shape when people get used to the other idea of Wilson namely universality perhaps in 1962 it was not so obvious but progressively people become used to the idea that after all what is important in the continuum limit is not exactly the bare model but to be in the right universality class. So perhaps we could have a simpler model based on only equilateral simplices which I will call equilateral regi-calculus although I have not seen very often these names I tend to use it and personally I learned this equilateral idea of regi-calculus from my friend François-David in the early 80s but I have put here two names but there are many other names so progressively it became a relatively mainstream in the case of triangulation of surfaces for which instead of using length we could just put equilateral triangles and then we could find a link with random matrix theory once this was... again myself once this was established it was natural that very quickly the same idea was proposed for higher dimensional triangulations and there are many names here and I just put the name of Ambien as symbolic of a group of people that proposed in the early 90s to use equilateral regi-calculus in any dimension by relating it to random tensor theory in order to quantize gravity so what is this... why is it that indeed it has anything to do equilateral regi-calculus with tensor or matrices by the way in dimension 2 it's because in the case of equilateral objects the regi-action simplifies you see because every equilateral simplex has the same volume you get just the number of simplices here instead of the volume and for the deficit angle it's so easy because all angles are the same in an equilateral object and therefore this is proportional to the number of faces that is to the deficit angle will be counted by a constant times the number of the length of faces which turn around d-2 dimensional inges and all the problem will be to define which turn around you know that for a surface if you glue triangles around the point the curvature is given if the triangle is equilateral by the number, the length of the circuit around the point if it's 6 like in hexagonal lattices you get that it's flat if it's 7 or it's 5 you have different signs of the curvature positive or negative and therefore the regi-action simplifies into counting simplices that is related to the cosmological constant and counting the length of around every d-2 inges so in the dual graph this becomes the number of vertices I will tell you a bit more about this duality between triangulation and graphs but just let's say that by duality you exchange d with p with d-p so this d-dimensional object becomes zero-dimensional object and the d-2 simplices here becomes just the number of faces of the dual graph which turn around these things hence the regi-action becomes a constant to the number of the vertices and a constant to the number of the faces for the dual graph that that's exactly what should be amplitudes of tensor models and the correspondences here these are continuous has to do with computing angles in an equilateral object but essentially that's the bridge between pondering random geometry with Einstein's Bert action and tensor models but then tensor models have problems because we will see that it's quite easy to define the vertices but it's much more difficult to define the faces first of all I have to tell you why a dual picture appears in particular when you want to place simplices by a field theory in which you have a fixed type of vertices if you have triangles the vertices here can be of many types they can have many different coordination but what has fixed coordination is a dual object the dual object is made by putting a vertex inside every triangle and putting dual lines there and then you see that you see only red vertices of coordination free that's therefore what we call a phi cube model more precisely because these triangles are in fact an embedding in the surface you find that these things are called by these three monographs and they correspond to amplitudes of matrices for instance in the case of triangulation you will find a duality between the partition function of triangulation and the one of matrix models with a trace of m-cube interaction where this m-cube is what generates this cubic red object but in the case of matrices there is something nice which is that the faces are easy to find they are canonical in Riemann graphs you do have faces and you follow the indices of the matrix and then in 1974 Toft found the one over an expansion based on this problem it is dominated by planar graphs he wrote this line which is probably the most cited line of his entire career but what was really great was to make a bridge between Feynman graphs and therefore something known in quantum field theory and the Euler genus here which is a geometrical notion in this way he created a powerful bridge between topology and physics okay so why is it that we get this in powers of a genus because we have rescaled appropriately this and therefore v and e give you essentially are fixed by this and this f when you get this combination this is 2-2g so you get dominance of the graphs which have Euler genus 0 then you have a story which continues to our understanding the continuum limit of this model okay but then when the same idea was tried in the 90s for tensor models it failed and the reason is there are several additional problems the first problem is there doesn't seem to be any good canonical definition of phases for d bigger than 3 you might have perhaps a definition but it's not very easy to work it out certainly it's difficult to enumerate the phases and perhaps in a way it's even a bit difficult to give them a geometrical meaning related to this very singular space is seen to dominate the corresponding sum so there was no simple clear analog of Toft 1 over an expansion so then in 2009 there was a real transition 1 over an expansion was found for tensors and it was discovered by a very simple idea which is to study unsymmetrized random tensors instead of symmetric ones I suppose that symmetric ones have to do with the idea of generalizing Hermitian matrices tensors in differential geometry and so on I think this is the reason that blocked people around symmetric tensors and in fact, symmetric tensors have a single un or on group for all their indices they are invariant under a small group which is changing all the basis by the same amount that's what you do in a change of coordinates for geometric tensors but it's not sure what you should do for algebraic tensors which are unsymmetric have a bigger sort of a bigger symmetry which is really tensorial because you could change basis for more for components independently essentially and then you get a canonical notion of faces in fact the full geomology you get 1 over an expansion which is not topological and you have less singular spaces in fact spheres which dominates the sum and then I sort of transfer track for this renewed approach to quantum gravity which now is equipped with a full 1 over an expansion so here I will stop just a minute for you to relax a bit and I will show some kind of a choke I will take some to show you a little bit this picture now that we have which is a bit of an echo to the one of horse and horse was doing in a different order so he was putting the s y k in the middle I will put it at the end so I will say that the natural sort of hierarchy is that you have vector you have matrices and then you have tensors and this is the question of rank so here you have rank 1 here you have rank 2 and here you could think you have rank 3 4 5 but my chance the one over an expansion seems to be really much the same as a common structure for any rank bigger or equal to 3 so maybe we will forever stay with three kinds of one over an expansion well not exactly because we will see later that within this world there could be plenty of one over an expansion also that's something that Valentin is a world expert on so let's explain here very quickly so here if you have vectors what are the leading term so what is surprising is the leading term we are going to see here you will have sort of chains of bubbles these are the famous objects that you find when you try to do BCS theory and all that here you have planar graphs so planar graphs are common in string theory and here you have the famous melons we are going to see this so here the melons look like a very tiny modification of this they look more like ladders like that than just bubbles but this is for the leading term so from the point of view of the leading term it's true that this guy looks in between of these two so it is simpler than planar but more complicated than chain of bubbles so in this sense you would be tempted to put this in the middle like Hoss was doing for SYK but from the logic of this it is not the case because that is just the n equals zero term but if you have a one over an expansion you should understand what it sums and here it sums over loops if you transform this thing into some if you collapse these things then adding loops is everything that you can know about in vector models you have no structure here no geometric structure more than adding loops here you have Riemann surfaces so if you if you go beyond the n equals the term you have the genus and here you will probe all geometries all piecewise linear structures in dimension d bigger or equal to 3 which is an enormous category so from this point of view over this leading term insert here clearly the one over an expansion itself is enormously more complicated here than here or here and therefore I will put this inclusion of course if you know about tensors you know everything about matrices and vectors but not the converse so for me it's a new chapter which is on this side and which is more general than the previous ones although we will see that the leading term is simpler then the question about physics that I don't know here you get physics you know of BCS or whatever here you get so maybe particles here are strings but here really is a question mark and I would tend to say that there is probably a kind of generalist notion so maybe we could call it brains but it is a coded word which is a bit special so I don't know but some perhaps more extended objects yes at this point I will stop my my choker and yes and listen to questions that's a very good question well I of course I hope so based on the mathematics the tensor will introduce two quantized gravity as a discretization bounded by the Einstein libertaction there are gravitons in this Einstein libertaction so why not in the quantum but we are now the thing is still much less developed and we will see that the contact with physics of this for the moment is very continuous and comes through this SYK model so we will have to explore the bulk and know much more about this but a priori certainly the people who introduced random tensor in the 90s to contact gravity and our our team also which continues on this track we hope to find gravitons and we hope to find perhaps there is a kind of unrealistic hope that we can also perhaps have formalism which is wider than just studying black holes or a DSCFT because a priori this program of quantizing random geometry is very general doesn't require a particular background it doesn't require objects like black holes it even doesn't require exactly a particular metric like ADS so black hole you need the weight rotation how do you define the weight rotation the weight rotation ok so here the question is first of all if there is no time or space we don't know exactly so the question of eventual emergence of space-time, euclidean temperature, time and space I have at this level not proposed anything yet about it and we can discuss it eventually but this was just this is just a broad classification very general no no n is the size of the tensor no I know on the gravity side oh yes so it's related to a scale so it is at the moment n on the SYK model has no physical interpretation and it's just a parameter you put it and you plug it to n equals infinity period but so in practice we like n to have maybe meaning as a kind of scale so in this sense it means that we would like to sort of join together so one over an expansion and some kind of let's say you probably limit in a certain sense this is not what is at the moment done in the SYK model but this is what we have in mind if we follow the root of sticking to the relationship which is much better understood between matrices and 2d quantum gravity when you go n to infinity you would like to have things called single or double scaling limits in which you see a continuous phase emerging but this continuous phase is then due to the proliferation of the simplices and therefore it is a bit like an ultraviolet limit yes is there any kind of physical limit where you see some graphs dominating of course so here yes the leading term is this physics is it semi-classical things that you can interpret some simple graphs that tessellate the smooth geometry well this graph tessellates spheres but they are stacked what mathematicians called stacked spheres so this suggests if you like when you see the melons growing they have a kind of tree structure but this tree structure if you interpret it in geometric terms in terms of triangulation is a bit like when you have a ball and you throw balls at it and it creates a kind of tree of balls which is still a ball topologically alright so this is a bit good for the point of view of aggregation so you have the problem of growth models also which could be interesting in this respect if you look at this melons from the geometric problem as a growth problem you find that it's called stacked triangulation of the sphere which is a very restricted family of triangulations of a sphere which is as people say locally constructable, exponentially bounded and so on and general triangulations of the sphere are much more complicated it's a but in the regional regio theory you could say that you could penalize local curvature so there should be a way to trace that back so that's a geometry which is maximal in the term of curvature with respect to volume did I say it correctly as well so what you want is exactly n to 0 instead of going to large general to see what you want you should go to any end so it's clear that there is a lot of room between the phase that naturally occurs here if we want to interpret it as a growing space and smooth for dimensional space which is flat it's exactly so my impression is that we have to understand eventually the scenarios of emergent geometry in many stages it's not going to this is very simple and what lies behind is very complicated whether here you have already a chunk which is relatively significant here the path is much longer if you want to go towards the flat geometries let me prepare a little bit perhaps the talk of Rasmaud which is going to be more centered by saying some details on the story of this tensor models so we have strange names they are so-called collared models and uncollared models so they create some confusion for the newcomer so let me say why it was historically discovered in a complicated way the first thing that was found was a wonderful expansion for what we now call the collared models and the Russians have found a poetic name they call it the rainbow model so I think I will adopt this poetic name so the rainbow model is simply to have D plus 1 different complex tensors of rank D and to branch them to branch their indices according to the pattern of the complete graph and therefore the complete graph with D plus 1 vertices has D D plus 1 over two edges an edge between every pair of summits and therefore the symmetry is really this one alright then you rescale the vertex with appropriate power of N and you find the Feynman graphs of these models are B-parted D plus 1 regular edge collared graphs and they are dual to the collared triangulations of orientable piecewise linear quasi-manifolds in dimension D so what is this category and so on I am going to give you a little bit of ideas but it is an enormous an enormous geometric category ok so that was the initial model this kind of rainbow model and for this model the one over an expansion was found by looking at the idea that if you have D regular collared vertices the graphs a bit look like that for instance this is a four collared a four regular collared graph there is no particular embedding it is not embedded graphs it is not embedded graphs in surfaces there is no surface at the beginning but there are several in fact surfaces because you know that you could write several and then you have a notion which is the notion of jacket it is the notion of a color cycle up to orientation so if you have four indices what is important if you have several, if you have because you know that you could write several cycles so for instance there are three important cycles up to orientation for four indices which can be defined as who is the guy in front of zero in the embedding here is the guy number two which one and three are left and right and in front is two here in front of zero I put the guy number three and here in front of zero I put the guy number one if I do this consistently for every vertex which I can do because I have these colors which I can only call in my craft I get as many different embeddings than there are different jackets these are the three different embeddings that I get some of them are planar, some of them are not in this case can you write the terms as polynomials in the tensor just to clarify yes I wrote this because I didn't write how you branch these guys for every vertex you have d guys but if you have the complete graph it's made of d plus one vertices linked with every pair so that's the idea for instance yeah I will do this for the pentagon because the pentagon is nice this is the complete graph the complete graph k4 yeah and it's even a symbol if you have a good Pythagorean you must be able to write this Eulerian graph without taking your chalk out of a blackboard which I didn't do okay alright this one is I will write it because this one is planar I will write it in planar this is k4 and so on and k3 is a triangle one is a triangle so in this case for instance you would have a t1, a t2, a t3 and a t4 and this guy has 3 indices 1, 2, 3, this guy also 3 indices so then you will branch every index with the index of same rank of every overturned source so maybe I can write for this a little bit differently yes there could be of course this could be a tensor with n1 times n2 times up to nd dimension you could have different dimensions for every pair in this case I took u of n to the d d plus 1 over 2 but you could take different dimensions for every pair alright so we have 4 tensors t1, t2, t3 so I will put their color index here and each of them is rank 3 so I will have an index identified here if I have a1, a2, a3 I will have an a1 at first sight here at second position here and at last position here because this guy sends a line to all the others now this t2 has a b1 I will call it if you like b2 if you like and you will find back this b2 b3 so for instance here I will have because I have the a1 here I will have the b2 shared with this guy and the b3 shared with so here I have to fill these things sorry and b4 you put b2 here I have to put b2 here I have to put b3 I suppose call it c3 or something I have put some bad names then you need c1 and here I need b1 and I put it no so this is one of the way and then I sum a1, a2, a3 b1, b2, b3 this is 6 sums because there are 6 edges in this thing and then if you want me to write this one there are 10 sums this is a simplex this is a simplex so this is a simplex in dimension 2 this is simplex in dimension 3 this is simplex of dimension 4 so this will be linked with simply show the composition of spaces in dimension 2, 3, 4 generally in dimension d you need d plus 1 point to create a simplex so I'm sorry I have perhaps lost everybody already but there are two equations I want to give you and these two equations they are miraculous because they are simple because the complete graph is simple and the idea is that you can count objects if they have a for instance I said it if you want to count cows and you know that you have two legs and then you just divide by 4 you find the number of cows it's not more complicated than that and the fact that the cow has always the same number of legs is because this geometry is based on simplices so in the case of the Euler relation for every jacket for every of these cycles you have an Euler relation that is you have 2 minus 2j which is the number of vertices the number of faces of edges the number of faces the number of vertices and the number of lines so that's this is the number of of faces this is the number of edges because I have d plus 1 over 2 fields for every vertex d plus 1 fields and every line costs 2 fields and this is d times v well I am now lost by this yeah this is because I I want to don't put the capital D on the first time that's a misprint this is a misprint sorry completely sorry so the important fact is this thing each pair belongs to factorial jackets because each pair is there whenever the colors are adjacent in the cycle and you count the number of cycles in which a pair of colors is adjacent you find this number so when you sum of the j you can find that this equation factorial d is the number of jackets factorial d over 2d multiplied by 2 minus 2 sum of the jacket of gg is a certain number of times v of ij appears in a fixed number of jackets therefore you can solve for f and you find this equation where omega is just the sum of the genus of the jackets the important thing was to find a formula for f because I remind you that 1 over an expansion has a term and to the power f so as soon as you have a formula for f you have an entrant ski to the n over an expansion so we now call this omega as a guru degree because it was we found it was not written before and there are so many degrees it's better to have a specific name and this object is not a topological invariant of the underlying manifold and then you find that the amplitude of the graph is given by this factor n to the d which is normal factor for a vacuum graph minus this and this is positive because hg is an integer or if you have an n model with non-orientable objects a half integer so the important thing is that there is this negative sign so you have just to find the leading term as omega equals 0 and all the others are suppressed progressively by a rule which is not according to the genus but according to this guru degree so we have then went on to analyze what are the leading sectors that is what are the graphs with n equals 0 and we found which have omega equals 0 and in this case we have an equation for f because remember if I take omega equals 0 f is just this so we have to analyze the graphs which have satisfied this equation and I will try but I think I am bad at this during seminars so I have tried also to put for you forward a second computation the second of my talk that we do need to so the first one was this omega computation of f and the second is a computation I want to show you because it was reproduced by Witten so it's now kind of famous so but you will see it's no more complicated it's perhaps even simpler so this one appeared in a paper with Valentin and Rassel and Aldo and then this is to to divide the faces according to their lengths you know you have the faces in this problem are always b-colored connected components of the graph so they always have even length and therefore you can say you have the face with 2 of length 2 of length 4 6 etc so it's a sum over s of the face of length 2s and you know that it has to be this but each face f has to have vf vertices and if you sum over s of the length times the number of faces you find the sum of vertices over all faces and therefore because each vertex again belongs to a fixed number of faces that's the number of corners if you like in the verse well you find the equation that the sum of s f2s is also constrained then you combine this equation and this equation in the very clever way that you take twice this one minus this one twice this one minus this one the sum of vertices in this problem will survive because it will have a factor 2 here and 1 here so it stays the factor f4 disappears because 2 minus 2 is 0 and all the others they become 2 minus s f2s for s bigger than 3 but this number you compute and you find 2d plus d d minus 3 v over 4 ok but this now you find 2d plus d over 3 you can pull it there and therefore you see that f2 is always 2d plus a positive number if d is bigger or equal to 3 of course this is wrong if d equals 2 that gives you the main difference between the matrix expansion which has not this bizarre property and the tensor expansion you always have short faces of length 2 because you have a short face of length 2 that's the starting point to pull the melons because by recursion after that it's relatively easy to prove that the structure is like this these things have plenty you know of faces of length 2 of course it has faces also of longer length but it has faces of length 2 and by using the fact that f2 is strictly positive you find that the graph which dominates here are simple they are the melons series parallel graph and we could also call them the superplanar graphs it's not surprising that they are more restricted than the planar graph because these are the ones which are planar in d factorial over 2 different ways canonical canonical and geometrically this will correspond to d factorial over 2 hegar decompositions which are given for free inside the structure ok so they can be enumerated and so on I will speed up because otherwise so I would like to tell you now another twist in the story which is a little bit technical it has to do with oncolog to some models so soon after the collared models were discovered we were searching for a better analog of matrix models because we say oh people are not going to buy it because they will not like the fact that I have d plus 1 different tensor they say usually I have a single matrix what happens when you have a single tensor and we said ok if we have a single tensor we should require not a un to the dd plus 1 over 2 invariance but just a un to the d invariance for each of the index of the single tensor and then we discovered that it was a bit like Russian dolls in fact the single tensor models were kind of restricted models of the rainbow model by requiring a certain kind of specific kind of vertices or bubbles for d colors out of d plus 1 so the basic objects are then the un to the d tensor invariance they are in one to one correspondence with regular d-edged collared connected b-parted graphs they are both the kind of vertices that generalize the traces of matrices and they are also observable for these models and they are dual to collared triangulations so the interesting things that they are both the interactions of wrong d tensors the observables of wrong d tensors and also the Feynman graphs of wrong d minus 1 tensor so this created a bit of confusion so I have a picture to try to show it to you so what is a tensor invariant well this is a tensor invariant this is another one so for instance here you have tensor invariance with 3 colors here I really put the colors this is better so these two are planars by the way these two are melanic this one is not planar and not melanic ok so this is so in fact these are all the graphs you can write with six vertices free black and free white which respect b-partedness and respect colorness so every vertex is regular color so you can enumerate them and if you look at how this changes with the rank for vectors you have a unique invariant it is a scalar product so unique connected of course you can do models in which you put this to a certain power but it's no longer connected it would be a kind of multi trace in your language ok this thing has exactly one single trace interaction for every number n you know trace of well this is written for these hard models mm cross to the n so there is only one for every value of n of course you can do non connected one so called multi trace models but if you look at the single trace if you like which we better would call single bubble objects at rank 3 or 4 then it rapidly grows so the tensor space is much much bigger than the matrix space the space of interactions for a tensor is much much much bigger than the space of interactions for of course for a vector but also for the matrix for instance v7 is when you have rank 3 you have 7 graphs these are the 7 graphs ok why 7 because you see 3 ok this one is there is only one because it's completely symmetric and these two come in 3 different types if you count that there is a special color so this is 3 plus 3 plus 1 is v7 and then you can compute all these things there must be 6 stick there is nothing 6 stick yeah there are the invariants at n equals 3 because I tried to show you as a function of the number because they are bipartite they have to have an even number of vertices which I call 2n and there I put you the numbers when n grows for instance this corresponds to n equals 3 which has 6 vertices and this is v7 and is there still the fact that you could have different dimensions for each index yes so here these are the vertices of a rank 3 tensor model because you have 3 colors if you would have rank 4 you would have and these will be like the bricks that you will use to create a simple decomposition in rank in this case in dimension 3 and if you have more colors in more dimensions so let me show you the next one which perhaps would be clear it means that a tensor model with u of n to the d invariant is a single tensor now this is a quadratic part which is canonical it's always contracting all indices of the tensor and here you put a certain number of vertices of this type for instance you can put this one with a certain coupling constant this one with another one and so on and then you get the most general single tensor model and its partition function and then why is it that rank 3 corresponds to graphs which are which have 4 colors well it's because you have the vertices which have 3 colors and then you have the big contractions which give you the Feynman lines between these vertices for which you can use another color in this case the green color so you see a tensor model like that realizes a sum a geometric sum over tetrahedronization of the space of any 3 dimensional space where are the tetrahedron well there is one thing which is a bit the tetrahedron are here you see there are 4 faces here and the tetrahedron are hidden here and in fact what you glue are bricks of tetrahedron which correspond to this so it's exactly like the generalization of a triangulation of a surface you have quadrangulations and so on hexagonalization this is the proportional generalization well I will skip this because I won't have it otherwise so just a word about associated field theories this has to do with joining together the n going to infinity limit and a kind of scale limit this is I won't say more but it is in relationship to Tansan model exactly like non-commutative field theories are a generalization of matrix models you have heard probably of non-commutative field theory as effective sectors of string theory they are not exactly matrix models because the propagator is not exactly one it has one over p squared well the same idea if you plug it into Tansan models you get so called Tansan field theories I think the important fact that I try to always stress to physicists is that very surprisingly these Tansan field theories are asymptotically free so this is something which was discovered in 2012 and I think it has not been sufficiently well probably we have to wait until Witten writes a paper about it okay so then there is a question of relationship of this to SYK so you know the story I will not repeat what Rosenos has explained very well so there is this question of trying to find a criterion for quantum chaos by looking at this 4-point correlator on the thermic circle the operators that you put there are not so important but what is important is that you can under very general assumptions of analyticity because these objects are sort of maximally spaced on the circle you can find an analyticity which is big a region of analyticity which is big and then you get a bound which is sort of optimal on the Lyapunov exponent for quantum chaos this thing and they argued also that if you see saturation of this bound this should be the indication of the gravitational dual but this was one of my questions with Rosenos so we have to maybe I don't understand fully this sentence but at least this is what they argued that saturation of this would be very interesting for the gravitational dual and so Kitayev found this model you have heard about which saturates this bound and which is solvable because of this and going to infinity limit which is due to the presence of this random tensor here which is quenched and in the infinity limit at this order not at this one but at this one you have in the somalons and they are related to the nice physical properties that excite theoretical physicists in this model so what happened as you know is that then there was a paper by Witten who stressed the link between this S.Y.K. model and our work by proposing to eliminate the quenched disorder of this single tensor by putting here many tensors exactly like in the rainbow model so the rainbow model was introduced by Razvan so I think it's appropriate indeed that some people have called this new class of models the guroviton models and here the pattern of contraction is this one for this tensor so you see they are the D plus one so it's somehow taking this propagator and equipping with this vertex you get this guroviton model which has the same one of red limit but which is also at least it was argued by Witten more satisfying as a fundamental theory because there's no quenched disorder so it looks more like a field theory and also he argued that tensor fields here are better for the bulk but I won't discuss this yes this is not important really not important but these are yes in this model in the model in the letter that he wrote he kept Majorana fermions here but after that for details it might be important yes but for the fact that the Mologne dominates you could as well take bosons or supersymmetric guys who took supersymmetric version of this and then the next step was well you have the rainbow what is the encolored so a step was done in this direction by Klebenov and Tarnopolsky very very soon after this one they propose to do this with a single tensor and this is based on a tensor model to the free tensor model found earlier by Karosa and Tarnasa alright then okay so many papers have appeared now on this subject and I would like to stress one result which is it's certain now that this group of SVK models are not the same certainly as soon as you go at the next order in 1 over N and I think this means they don't really describe probably the same physics and then I would like to I'm sorry I missed yesterday the talk of Frank so I would like to show another line of research which was starting from string theory and which connected with tensor model because in order to analyze sort of black holes and their translation property at large dimension D because black holes are made by piling N brains you have to find a kind of large D limit of matrix models and Frank was searching for it is he there yes so you will go at me if I and then he was searching to a D expansion which will select non-trivial subset of graphs within this and in fact he found that the most natural thing was to use tensor interactions and scaling of his Karoza-Tanaza model at least in the simplest case and that these were the superplanar graphs inside this which would govern the 1 over D limit inside the planar graphs and so on and I think this is a fascinating new research direction okay so now I will try to finish so I have not time to tell you about the geometric more about the geometric aspect of tensor but let me say the following the fact is that there are now many models with the interesting SYK physics but there is one thing just common to all this model you need a random tensor without random tensor either quenched anneal, rainbow or whatever without tensor no SYK physics conclusion well I like this sentence random tensor pulls the strings so if you have just 5 words to retain from my talk I would say random tensor pulls the strings okay I didn't say push the string I say pull the string okay it's not very aggressive sentence but behind the physics of the SYK model after all we are not surprised because they were introduced and studied since 25 in order to discretize and quantize gravity and now it means from this fact and this conclusion stands out a program and this program is understand the connection between holographic and geometric content of random tensor and I think this is a challenge and I will stop here can you please elaborate a little more on the physical differences what the 1 over n expansion brings between the SYK and the tensor model no I think that would be I would prefer that directly for instance Valentine who was one of the author of this paper answers this question perhaps but okay could you say of course the physics there are some technical problems one of the key aspects is that you find similar graphs but they do not appear of the same order in the 1 over expansions does it change anything important results we haven't calculated amplitudes at all so our results are purely combinatorial so we just identify which graphs contribute to which one I have maybe a comment on this very interesting question I think the fact that the corrections to the leading order can be very different in tensor models and also in these large D matrix models comes from the fact that you don't have the formalism of the auxiliary field that you have in the simplest SYK case when you can formulate everything in terms of this auxiliary field by expanding around the saddle point you can actually systematically compute all the 1 over n corrections in the tensor models or the large D matrix models you do not have this formalism and that indicates that the corrections to the leading order are much more subtle you have more graphs it's a more complicated business for the corrections even though the leading order are very similar have you tried to put matter on this? not really not yet there are several things which have kept us busy and we need absolutely to interact with other groups for instance we have a tendency to be mathematical physicists and therefore we need also input from physics this SYK is a very good occasion to do that a sense of matrix models where you put eyes on models there are dimer models we have done some of it and also we have done some application to, you were also involved to gas physics but I think there is plenty to be discovered and in particular I would like very much to probe the growth properties of tensor models compared to the growth property of triangulation and matrix models we know this is linked to for instance the and also the tracy-widom statistics and all that so there is an enormous field I think in the statistical mechanics to be investigated on that because in fact there are some similarities with the growth of matrices and some differences I think this is a very important thing I have a comment on this question what it said on SYK so if one compute the length of the exponent for a block called the Einstein gravity if one gets this value that being voted to pi times the temperature the maximum value if one computes it in string theory then it is no longer the maximum it is below the maximum where the corrections are set by the string scale so the two types of insight had been that since the block hole has lay-up effects for two pi times the temperature we have a model of block hole which would be two pi times the temperature by model of block hole what was meant was block hole in Einstein gravity not for instance I mean you could say you have a clue if you took a block hole in string theory and made the string scale very low then you don't really want to model that you want to model in Einstein gravity or string theory with a very large yeah so that is almost Einstein gravity could I have hoped that this would be so this is clearly a necessary condition a sufficient condition and that turned out to be false because the bulk deal of SWK has the analog of gravity in two dimensions the dilaton but then there is as I talked about a tower of massive particles which don't decouple and they are light so it is analogous to having a string scale which is a water radius scale which is very far from from the gravity, pure gravity and is it a question if you want to include the cosmological constant as you need for instance do you need two kinds of tensors I need two parameters one is the coupling constant in this model and the other is the scale N and then I need to go to a critical regime otherwise the tensors are the same just a single rank oh the rank I don't need to change the rank a priori but there is a very interesting possibility now that this rank can grow to infinity which in a way would correspond to look at a kind of mean field theory of gravity but you don't have yet this critical point to be found either Einstein or Einstein which goes to a critical constant so for now it's not known a critical point there is a critical point at any rank for the Mellon series yeah it does not go to Einstein oh no, what you mean by it doesn't go to Einstein well geometry well yes if you interpret immediately the geometry has restricted to this kind of sphere and if you take as metric the graph metric if you do these assumptions then indeed so because you know there is a question of tensors but they represent geometric gluings but do they represent if you take then the point of view of equilateral radial calculus then indeed you will fix sort of a matrix which will be the graph metric and then you have to see what is this random space which is infinite sum of all Mellons at critical point this is indeed then what is called Aldous tree so this certainly doesn't look like close to any close state of dimensions the rank of the tensor so it means that in order to get to a smoother phase in this interpretation you absolutely need to include going to critical points governed by fluctuation around this one over a leading term so it means that you have absolutely to enter into the one over an expansion well then this becomes really difficult more difficult certainly than just double scaling that's