 let me start first with personal recollection happy birthday of course Tiwo also i don't know when it was second thing you see we wrote with Tiwo couple of papers moreover one of the papers is incredibly cited it's called K-influccion І я можу тебе сказати щось про історію, як ці пейпо, кай-інфлейціон, епід. Тібо був шеріг, офіс, завжди, знову, з якою, більше ехс, вулиць, і, окей, дуже гарний док, був приїжджений до семіналів, спілки, і з того, коли співер був вонтані, на різній різній лоджіх, Док був виразом його, і був, гав гав. Так, це був вийшовий критері. Нормально, я був, коли я був визит, в АТС, я розумію, це був один вечір, коли все це був піет, я був сидіти в випадку, Від мене був док, і я був ділянним у два часу, всі в Бюрсереві діло, тому що нормалі, як север. У цьому сіті є ті, є лише один кінць, який ще працює. Манапрі. Я був думав, що я буду в Манапрі, і спинтати добре життя з цим ботелом. Тіболу, звісно, завжди хоче робити щось. Перед часом, як він почав дістатися, Тібол вже прийшов калкуляції. Спори, якраз, плежа, of thinking that maybe idea is not wrong. It came to the title to our paper. At the beginning I said, kinetic inflation, but then I discovered that somebody already spoiled the name. Then I thought, k-inflation. Тібол told me, nobody will publish paper with this kind of title. I told him, let's try. At this point, Ulis said, gaf, gaf, and then it was decided. So, in principle, it should be on the paper. But because the paper collected incredible number of references, which I would never expect, although it's not my best paper, it's not the best paper of the book, as far as I understand. Then we had also with Тібол some discussions about interpretation of quantum mechanics. And Тібол completely agreed with me that there is only one interpretation of quantum mechanics. I will not say which one, okay, not just to damage our reputation. But then I told, Тібол, let's write some book for physicists. Тібол said, for physicists, it's completely useless to write such a book. And this is the reason why Тібол publishes this book. And I have made it with my picture here. The only thing actually you cannot recognize me is a tie, which I never care. I think it was the right movement from the side of Тібол, because physicists' community so stop that they cannot take anything new, if they were somehow in the darkness for 40 or 50 years. Now, the topic of today's talk is discrete gravity, some paper, which we wrote together with Alie Shamsidin during last months. And at the beginning it was supposed to be somehow generalization of, how now to go, wait a minute, I think wrong. Okay, we were thinking about generalization of latest gauge theory to the case of gravity. Okay, two words, reminding or recalling to the people what does it mean, latest gauge theory. Of course, it was an invitation by Wilson. It's not just discretization, usual Young Mills theory, it's something else, but in the appropriate limit it gives you Young Mills theory. So what you do, you make some lattice in the space, which is very definite lattice, namely you use decarve coordinate system. Everything what I will be talking about is an Euclidean space, but you can make weak rotations, because the problem. And then take on this lattice a set of plackets, which I wrote here. And then with each point, which connects to vertices, with each line, you associate some unitary matrix. For instance, UN or SUR and write it as exponent of some parameter A, which characterize the size of this placket. And A mu, where A mu, after that becomes gauge fields. Point X here is the value of what will become gauge field in the middle somewhere of this line, which connects to vertices. You see, and mu index is the appropriate vector, which connects to appropriate vertices. And after that you build this kind of combination related with the placket. So it's a real part of trace of the products of the appropriate matrices when you go along this placket. And it's clear that the series, which you are getting, is invariant under local UN transformation. Because with each vertex you can associate some unitary matrix. And then when you are making these transformations, this object remains invariant, as you understand, because appropriate matrices, unitary matrices, just cancel each other. And after that from this object you build the series, which has so-called redundancy with respect to an arbitrary gauge transformation. I do not know what was the physical motivation for this thing. Of course you could use it on computers like people are using it and claiming that they got some result in SU3 or whatsoever, but nevertheless, for instance concerning confinement, etc. When you go to continuous limit, you know that you are not supposed to trace this result. Now then we started to think that maybe we could make also lattice gravity, of this kind of lattice gauge theory and there is much more motivation to think about it. Why? Because as I think everybody would agree, we cannot check the scale smaller than the Planckian scale. There are several reasons for this. First of all, if you will take Einstein's theory and quantize it around Minkowski's background, then you will find that quantum fluctuation of the metric on the scale smaller than the Planckian scale becomes much larger than unity. Therefore there is no any point to speak about the background Minkowski's face there. There is Pamiranchuk as far as I remember was saying in the Planckian scale we have liquid or boiling liquid of operators. I do not know what kind of sense of it, but sounds nice, right? Second reason why you cannot check scales which are smaller than the Planckian scale. Because from the point of your physics to check some scales you have to collide particles with the appropriate energies, ok? Which would correspond to the wavelengths smaller than the Planckian wavelengths. But this would mean in the center of mass coordinate system that the total energy should be larger than the Planck mass and in this case of course if interaction will be efficient you would form black hole with radius which is larger than the Planckian scale, right? So in some sense there is no way actually how you can verify this smaller scale. Also when we go to quantum field series that we know that there is infinite number degrees of freedom and one of the infinities which is more relevant for this number degrees of freedom is related with deep ultraviolet regime, you see? And as a result there appears all this kind of quite about super Planckian problem for cosmological perturbations and so and so and so. I don't understand actually how could you treat even the things in the scale smaller than the Planckian scale and write dispersion relations there. It's impossible. Therefore, of course, much more reasonable it would be to have finite number degrees of freedom per each volume, let's say volume of the Planckian scale. It would be much more logical and then when for instance, the universe is expanding the number degrees of freedom or the number of cells will be growing. There is nothing wrong about the change of the number degrees of freedom. If I don't take pencil we'll break it into pieces. I got more degrees of freedom as you understood. Gravity also can do this job. For instance, in inflation in this case you would have multiplication of the number degrees of freedom of the number of cells would be just growing of elementary cells but the main idea is to have just some kind of elementary cells which would be indistinguishable inside and with each cell associate for instance only one value for the scalar field if you consider scalar field and then there appear immediately some obstacles because in the latest Gage series most of the people I think nearly all of this I didn't go too much in the literature use this kind of Descartes coordinate system rectangular cells but we know that in general relativity you can take any coordinate system and in principle it's in your hand how to divide the space into elementary cells then you can tell that take some kind of thing and divide it into elementary cells some kind of coordinate coverage like this one and then associate with each cell one number and there can be it's very similar to what is going on for instance in the case when we have quantum mechanics because as you know in classical series in phase space per each cell you have infinite number degrees of freedom but in quantum mechanics per each elementary cell you have just one degree of freedom roughly speaking of finite number degrees of freedom finite number of states in quantum in classical series infinite number of states in each finite volume in quantum series it's not the case but then the first idea would be just to take this kind of subdivision and numerate all these cells by integer without thinking about any internal structure but in this case there appears immediately question if you just discretize your space why is the point what is the dimension of this space because when space consists of the finite numbers the dimension is all finite points this dimension is always equal to zero therefore nevertheless we still have to think about cells which have no internal structure and for instance in continuous limit the dimension is defined like the dimension many fold of the classical space for which each point has a neighbor foot a homeomorphic tool if clearly that d-dimensional space and in discrete space we could use this kind of definition which I gave here with which mathematicians perhaps can be unhappy but they can refine in discrete space we define the dimension d assuming that each cell has 2d neighboring cells that share a common boundary with which individual cell for instance if I will take 2 dimensional case because it's much easier to draw and will take this cell of elementary volume then it's surrounded by 4 points and then in d-dimensional case I can numerate all these points by integers by d-integers which I can combine with this n-alpha for instance if this cell I will numerate by numbers n1 and n2 to dimension n1 plus 1 nearby cell here is nearby cell here n2 plus 1 is nearby cell here and these cells are approximately here now when we have the structure of cells we could think about further structure for instance I could define function but in this case it would be discrete function for instance it would be some scalar which takes one value in cell n and takes a different value in cell n plus 1 n plus 1 can be any cell here I will skip this alpha because formulas will be too messy otherwise so in some senses I told you in this case we are introducing finite number degrees of freedom per elementary cell like in quantum mechanics although it's not quantum mechanics next thing which we could define are so called shift operators which bring us from one cell to the other cell these operators for instance if we define already function and these operators are associated with each cell as a result of action produce the value of the function in the nearby cell which has common boundary with the others then these shift operators in the d-dimensional case form a linear space because you could define multiplication by number of some of the operators according to this formula you can also define inverse shift operator just moving to the cell which is on the opposite side and alpha shows you direction here in which you have to move ok, so far so good next step we will need also to introduce here so called tangent operators because after that we will need to get continuous limit and these tangent operators should become like tangent vector to coordinate lines because we do not want to lose these so called difumorphism invariance and they are built within the given cell as a linear combination of shift operator and inverse shift operator for instance in two-dimensional case C2 makes you different of the functions in two nearby cells divided by 2 you see so now next step would be to introduce of course scalar product and you could find perhaps I needed to write with this formula you can find a linear combination of these tangent operators with the coefficient which are inverse what is called soldering form which is just decomposition for instance of tangent operator in terms of this v special vectors which are orthogonal and which form a basis okay orthogonal basis at each point and the series as we will claim should be invariant with respect to the group of rotations okay which preserves this scalar product and these vectors which are orthogonal orthonormal people normally call field bias or infodimensions field bias so with which cell we will associate the appropriate the corresponding field bias and we want to build the symmetry the series with the local symmetry group which will be the group of rotations SOD in each cell so okay under this rotation which in the Minkowski case would be just Lorentz transformation and would respect the redundancy related with the velocity of refollin-Einstein elevator right okay under this rotation the set of fear-bind v goes to the set of fear-binds v tilde okay and the angle of rotation of course continuous symmetry the angle of rotation is determined here then the next step we should be done here similar how it's done in general relativity is to define the notion of parallel transport if you would take this tangent operators you would be in some trouble because normally after that when you would be defining for instance Riemann curvature as a result of commutate of two derivative of two vectors you have to keep in mind that there is no lagness rule in the discretized cases this is the reason why to determine the symmetry real gauge symmetry one has to use always this tangent group of this fear-binds because this local group in fact determines the group of redundancy it's not defi-amorphism defi-amorphism is just renumeration of the point not more but this redundancy reduce the number degrees of freedom to the right number degrees of freedom for instance in general relativity and when I am taking for instance one of the vectors from fear-bind plus one b and bring it to nearby point then I will define the rule of parallel transport union so called spin connection group elements not spin connection algebra because you have to understand why we have to have group elements here because we have to exponentiate these things since its finite shifts and therefore ok, we could define the element of spin connection algebra which give you parallel transport using this formula where omega normal spin connection which we know is contracted with the generators of the rotation group in the vector representation here after that it's very easy to find the parallel transport for the tangent operators because you just use the fact that the scalar product of fear-bind with the tangent operators is scalar it's not changing when you make parallel transport and then you can express all affine connections in terms of spin connections and appropriate ordering form Hello? Yeah So this formula can allow us entirely express gamma affine connection in terms of spin connections I mean elements of the algebra here this omega and appropriate fear-binds soldering form at the point n in the cell n plus nearby points So, of course this crystal symbol do not contain any new information because parallel transport is already determined by spin connection then if we will require that there is no torsion then we are getting this kind of equation which allows us to express on spin connections just in terms of soldering form in n and nearby points Now, as I told you the series should be invariant with respect to local S or D transformations and if I look how elements of the spin algebra are transforming under this group I see that they are not transforming covariantly, it's the same story at least when you write transformation law for the gauge field and there appears extra piece with the derivative so it's reflected in the fact that here earth is taken in the in the nearby point but if I will build, for instance from these elements of the spin connection algebra and shift operator this kind of operator then one can verify that this object is transforming covariant this is the reason why if I will consider this kind of expression which is built out of spin connection elements of the algebra these are alpha-beta at point N is transforming also covariantly this is the reason why after that I can make gauge invariant combinations and moreover this kind of expression respects this redundancy related with S or N group if you would look what is this how it's built sorry then ok, what we are doing we are taking also some kind of placket which connects different cells and go along this placket anti-clockwise and then subtract appropriate quantity which corresponds to clockwise now автоматically we are getting the object which is of course when we extract these two things which is antisymmetric in alpha and beta so for S or 3 S or 2 and S or 4 groups our alpha-beta could be rewritten in this more familiar form where a spin connection curvature G are the appropriate generators of the rotation group doesn't matter in vector representation or spinner representation and after that we take these elements contract appropriate indices and we are building scalar curvature in this discretized gravity and therefore we can write after that immediately action just summing overall end there is a factor here weight factor which we were able to determine considering hermity of derecooperator only I don't know how to get it differently which satisfies this equation and in the continuous limit of the determinant of the soldier in form uc2 root of the determinant of G now equations are very complicated which we are getting because in abstract form it's okay but already when I try to apply or to write this equation explicitly in two-dimensional case when all cells are numerated just in one and then two and I will use the freedom in the using rotation group actually freedom is the choice of cells to set all fear binds to be equal to each other because here there is only one fear bind then first of all I could rewrite spin connection not in exponential form but in the form because it's s for two group to matrix okay in the finite form and we can resolve the equation for the absence of the torsion and as you see we are getting rather messy expression for the spin connections out of which we can build for instance this two-dimensional scalar curvature when we substitute this thing here therefore it's a lot of job to extract equations I don't speak about solving them but the things is here ideological from which you could start at least quantizing theory after that with the idea of having elementary quanta of the space now the theory which actually five minutes I will finish in five minutes because as you see transparency will come to the end of course it's very non-trivial to see that you will get correct continuous limit we spent quite a lot of time with it because if you would for instance take some coordinate x multiplied by epsilon and then after that epsilon you will send to zero that all point will come to the same point therefore if you want to get continuous limit you simultaneously has to take this parameter epsilon which determines you the typical size of the scale send it to zero but numbers which you use to number the cells should go to infinity it's like shown here you take just straight line you take particular point of the manifold you are discretizing the whole thing then this point corresponds to n equals 3 but when you will decrease the discretization then n will become 5 so now keep the point x but then take limit epsilon goes to zero then in this case the shift operators are determined like this one where epsilon shows you each cell moves you shift operator your function f from the cell numerated x and then when you take limit epsilon goes to zero then you see that this tangent vector when it's applied to some function f I think there should be f here gives you nothing then just derivative a long appropriate coordinate line you see so we got this thing and all these kind of shift operators became just tangential operators now when we are defining parallel transport for the fear binds of course epsilon should also enter here and then when we will take into account appropriate modification of our definitions this epsilon non equal one then we are getting well known expression for the spin connection curvature and the action which we have here which was some overall cells become just normal Einstein action so as I told you some kind of scheme would replace latest gauge theory but what to do considering concrete calculations is actually the next step and also all these kind of considerations is applicable in any number of dimensions but in 2, 3 or 4 dimensions it's all these exponents can be written down as matrices in other cases it cannot be done 5, 6, 7 dimensions I will survive with 2 and 3 dimensions because I will think that the things would be very useful if we will use Hamiltonian approach to general relativity and then at each constant time hyperservice we discretize the space in terms of the cells of the finite sides and then the theory of the invariant with respect to this rotation group which can be promoted to 3-dimensional rotation group if you want to go to 4 dimensions then the whole thing can be promoted to the local for instance local Lawrence group ok, so thank you very much thank you, are there questions? I have a question did you hear me? yes from the beginning you consider a plane with a lattice of square and the number of neighbors cells are 4 but suppose I consider the plane with triangles or with a pentagon and pamoson something non-symmetric or in 3D with a platonic solid what happens to your theory? this is a connection between the number of neighbors or maybe reflected in no, you are not allowed to do because you can make as many neighborhoods as you want but ok, when we are defining for instance dimension of the space we are assuming that each cell has appropriate and the same number of 2D nearby cells which share the common boundary you see when here you are not assuming some kind of internal structure because you don't know how to define the length, how to define the straight line it's very different from the logistics because the logistics assumes some kind of subtraction here it's not the case here we do not know even what are the lines that you could draw it like I was doing it before here it's very cute but of course in principle you could make here more cells surrounding this cell but then you have to repeat this procedure for everything you see and the only justification that it works, it gives you actually very straight forward continuous limit respect the symmetry for the continuous manifold I mean symmetry with the local tangent group yeah and it was not so easy to obtain because you understand that a lot of the notions of differential geometry which you have you cannot bring to the case of discretized space and one of the headache with it is that there is no labnis rule okay in the case when you are discretizing the things after discretization you see otherwise I could survive for instance with a fine connection but as I told you a fine connection in this case do not allow you to respect this kind of redundancy related with physical freedom of taking Einstein elevators at the same point but with different velocities but with the same acceleration of course you see this is actually the physical reason if you would use just free the momentum rate it doesn't bring you too much okay, thank you I see there is a question by Nikita Nekrosov online maybe we can make this a quick question yes hi the question is how does the topology change with respect to spaces of different topology in general without topologies I have no idea we didn't think yet about it but perhaps it could be also to generalize to the case of different topologies that would probably require changing the number of neighbors in the simplest case you know that when for instance we are speaking in the dimensional space even in continuous limit we are not speaking even about topology we are speaking about neighboring which are isomorphic to the efflidian space of the appropriate dimension here you see I do not know how for instance right away to do actually the same topological and trivial stuff thank you but he just he just was the first step after that I told you applications but you know that it's not actually equations of course is infinite differences as you have seen even in two dimensions they look I would say not very straightforward like this one in two dimensions the special coordinate system as you understand so this is actually what you don't need to claim that we solve all the problems right away you see but nevertheless here the idea of having finite number degrees of freedom to a plant scale is realized somehow and there is no trouble with it you see because as I told you you can have finite number degrees of freedom to a plant scale but number degrees of freedom can be growing this time so maybe in view of the time let's thank Slava again . . . . . . . . . . . . .