 Right, so first of all, let me thank the organizers for the opportunity to give this talk That's the title and the sub title could have been and now for something completely different Not because it is about quantum gravity. You have heard a few talks about the asymptotic Safety scenario for quantum gravity But because in this talk there is no space time And there will not be any space time until towards the end In fact the goal of this Formalism and of other non-perturbative quantum gravity approaches Is to provide the candidate theories candidate models for the microstructure of space time From which a space time should emerge in some approximation and in some regime of the models I'm going to introduce for you the models. I will be dealing with very briefly and only For those points for those features of the models that you will need in the rest Then I will highlight what the problem of the continuum limit is in this formalism Give you an overview of how the function realization group Techniques have been applied recently to tackle the problem of the continuum limit in these models and Towards the end you will see something like a space time because I'll show you how we are Extracting an effective cosmological dynamics for very simple space times homogeneous spatial geometries From the formalism in one particular phase of the theory Let me start I introduce for you the formalism the formalism is called group field theory and It's a class of field theories and you should understand them intuitively as Quantum field theory of space time and therefore I stress once more the fields Appearing as fundamental variables in this formalism are not defined on any space time a space time manifold Is there to be understood intuitively as quantum field theories for the building blocks of a space So slightly more technically they are field theories over a group manifold actually several copies of a group manifold And I'm dealing with with complex fields the corresponding phase space From which you take your data Would be this code several copies of a cotangent bundle of a group manifold The reason why these are interesting for a quantum gravity Models is that you can interpret this phase space in the appropriate dimension and with the appropriate choice of the gauge groups of groups as The phase space for discrete simplices. This is a four-dimensional example. So D equal four. So the number of copies Here is the number of Space-time dimension for the space time you want to reconstruct in some regime in some corner of the theory and in models That are related to quantum graph quantum gravity. Sorry The group is chosen to be the local gauge group of gravity. That is the Lawrence group in in four dimensions and Lorentzian signature in D equal four you are going therefore to have a complex field that taking value from four copies of the group So the function of four group elements if you go to the cotangent space something like a momentum space There will be four le algebra elements And you can interpret them. I'm not going to details here But the intuitive picture is that this four le algebra elements can be understood or put in correspondence with four area by vectors For the four triangles on the boundary of a tetradron, which is a fundamental building block for three-dimensional discrete Spaces so the this means that the Being a field theory you start from a folk vacuum which represents a state with no Structure no topological no geometry structure. You stop building up your states And a single quantum of the field can be understood pictorially as such single tetrahedron Labeled by the data I showed you in the previous slide In a generic quantum state is going to miss going to be some arbitrary collection of such tetrahedra Including those configurations when they are glued together. So you form an extended structure of a discrete Topological type so a simplicial complex. There is a dual picture instead of Drawing a single quantum as a tetrahedron You you represent it as a vertex node with four outgoing links labeled by the same group theoretic data and Extended configurations therefore correspond to graphs I'll come back To this in a in a couple of slides So you give you specify the dynamics of the field theory by defining some classical action with the quadratic part and some higher-order interaction part and the key point is that The arguments of the fields appearing in the action are Paired are combined in the action that are convoluted by group integrals in a non-local fashion Then we show you in what sense Again, let me stick to the D equal for case and to the simplicial setting Each field corresponds to a tetrahedron the interaction can be chosen to represent Five such tetrahedra They are glued together to form a force in Plex, which is a basic building block for a four-dimensional discrete space Take five Fields each with four arguments and you convolute them with such pair wise Identification to mimic the combinatorics of the five tetrahedra that form a force in Plex So just like a single quantum of the field is a building block of space a Single interaction vertex like this is a building block of a discrete space time This becomes apparent in the perturbative expansion of the quantum theory around the trivial vacuum Because when you expand in Feynman diagrams each Feynman diagram by construction Would be a stranded diagram of the type I showed you in the previous slide due well to some cellular complex So you're defining the quantum theory by summing Over all possible complexes that you can generate by gluing together the cells Dual to the interaction vertices the Feynman ampoules of course are model dependent But rather generically they can be understood as a discrete gravity path integrals With gravity discretized on the simplicial complex dual to the given Feynman diagram So you're trying to define a quantum theory of gravity if you have chosen the appropriate model with appropriate data with this type of discrete geometric interpretation as A sum over Transulations weighted by a discrete gravity path integral So these are exactly the two main ways of trying to define lattice quantum gravity Dynamical triangulations on one end and quantum regi calculus on the other hand Another way to represent the same Feynman amplitudes for any given model is as Histories of spin networks where spin networks are exactly these graphs labeled by group theoretic data that I showed you earlier Which are the states of such field theory and they are also the sum of a histories formulation of loop on to gravity in fact, you can see loop on to gravity as a sort of You can see group field theories as a second quantized reformulation of loop on to gravity in the very general sense that The states of quantum gravity or loop on to gravity appear as particular many body states in a group field theory formalism And there's no point in going to details, but here. I just wanted to highlight therefore that You bring together ideas and structures of us from several approaches together within a quantum field theory formalism The reason why this is useful is that as I'm going to discuss in all the rest of the talk This quantum field theory methods are crucial To deal with large number of the discrete degrees of freedom that are at the root of all these approaches and The problem of dealing with the large number of degrees of freedom sector or the formalism is indeed the main Open problem in all these approaches So here is just propaganda So we're just group theory as the as a cross-road in the sense that I try to highlight of several formalism and ideas Of course is a cross-road you may decide that you you're going the wrong direction and come back from Where you were coming from either from loop on to gravity or tensor models and dynamical triangulations or Lattice quantum gravity, but it remains the fact Okay, the open issues that I mentioned for quantum gravity are many, but the one and all of them acquire a different Interpretation then came and can be tackled in a very fruitful way. Thanks to this field heuristic formalism The one I will focus on is the continuum limit which I interpret here to be the task of Controlling the quantum dynamics or more and more of the interacting degrees of freedom that this formalism these approaches are based on So what is really the problem for group elements? Yes, there is the arm measure on the on the group Yeah Sure from the from the formal point of view Yes, if you were dealing with this complex scalar field on the sphere It would be a group field theory on SU2 because the tree sphere is SU2 The extra peculiarity that it was not in your summary and I'll try to emphasize it again It's the fact that the interactions are not local on the group Meaning that the various arguments of the field or the various fields that appear in the interactions are not all Identified locally meaning you don't just assume that the all interaction takes place at one point on the group manifold But you do not yeah, but do not yeah, but even Well discreet. It's a league group. So there are continuous variables, but Yes, but you you identify in the interactions For example one group element from one field interaction with one group element in another field in interaction Not all four of them Which is what you would do in a in a local field theory Apart from this the summary is perfect So the problem of the continuum is that we are starting from degrees of freedom that as I trying to emphasize a priori do not have any interpretation in terms of smooth manifolds or smooth metric or matter fields and you want to You want to get to a Space-time in particular you have a direction to explore which is the one of increasing the number of degrees of freedom and I want to emphasize that This Direction is independent from the classical to quantum Limit or approximation The reason why emphasize that is that we know a lot in in all these quantum gravity approaches About the classical limit of stays and histories where few of the initial degrees of freedom are involved So where you stay at the discrete level that is more or less under control What is not under control is this continuum limit? Okay, the non-perturbative sector of the theory if you want Okay, and the tool the crucial tool to study this Limit or more and more degrees of freedom is as you know much better than I do the renormalization group So what we do expect in general is that The renormalization group will allow you to study the collective behavior of such interacting degrees of freedom And in general we because they're quantum and interactive we should expect that When we increase the number of such degrees of freedom that we deal with we are going to find different phases separated by phase transitions part of the Problem here is to study the phase diagram of the corresponding theory the corresponding field theory the group field theory And again to guide your intuition. We're yeah, yeah, we can use it also the discrete level for perturbed Renormalization, but yes, we are studying I'll show you in the next few slides how we use the renormalization group so far The you should think of from the point of view of the problem or reconstructing a space time The intuition we have is that we should deal with this group field theories as the analog of the atomic quantum field theories in a condensed Matter system. I gave you the quantum field theory for the atoms and I ask you okay Extract the hydrodynamics extract the macroscopic phase diagram. Tell me if you get a super fluidity or the like So this is what we try to do and This goes this work We are doing in the group field theory setting goes in parallel with a lot of words that is going on in this related approaches in loop on to gravity spin forms and so on where indeed we know about different phases of The corresponding quantum theories although not yet for many of them at the dynamical level so let me go to the Exact to the use of the functionalization group is the idea is to be naive meaning that Let's take seriously the fact that we are dealing with quantum field theories for a second Let's forget about the absence of a space time the interpretation that is quite exotic of the quantum states and of The effective physics you want to get to let's forget about that Let's treat them as standard ordinary quantum field theories on a group manifold So we use the group structures the killing form the topology and so on to define a notion of scale So in practice we do what you would do in now in standard field here You expand in modes where the modes are the eigenvalues of the Laplacian for example on that manifold and That defines for you a state a scale the subtleties will come at the con in the context of the Interpretation, but formally you can just keep going you index the scales by group representations And you set up the The standard enumeration group analysis The main difficulty and that's why I emphasized earlier that that was a key difference from local field theory is to keep track of the complicated combinatorics of the diagrams of the Feynman diagrams of these field theories at the perturbative level and But more generally of the interactions that can enter the possible interactions that you can have in a gft action at the non perturbative level and That forces you to redefine or adopt the several notions from field theory from standard field theory local field theory So we are doing that the cluster models where we know more Are those where we have a sort of an analog of a locality principle? Which is the so-called tensor invariance or tensor reality as they call it so you define a cluster models in which an infinite set of possible interactions are labeled are in one-to-one correspondence with Bipartite decolored graphs, so if you have a complex field you you associate a field with the white node a complex field with the Black node that you color the arguments of the field and you force yourself to only glue along same colors This gives you bipartite color graphs. You can classify all the possible contractions of fields Corresponding to such graphs and that gives you the theory space it's infinite it's large enough, but it's characterized by this sort of Symmetry principle and then you choose as kinetic term Laplacian on the group manifold Which is consistent with the way we're defining scales Okay, this is a typical Feynman diagram I don't discuss but each of these bubbles is a possible interaction vertex That's where the non locality of the interactions comes in. You see you don't get a node with one two three four lines Coming out, but you get something with some internal complicated structure Okay, so From this point onwards once you understood what is the key technical difficulty is field theory meaning You just Set up as I will do next in the next slide the standard function renormalization group analysis in terms of the veterich equation And you compute the beta functions whenever you can just Well, I'll skip this in case somebody asked I'll say more about the geometry interpretation of these scales and of this flow But let me keep going here. So at the perturbative level. I just Mentioned that there is a lot of work. We know a lot of the renormalizable models to all orders in perturbation theory We know just renormalizable models super renormalizable models. And of course, we know not renormalizable models as well We are we are getting quite a lot of experience. What seems to be generic is that we find asymptotic freedom And whenever we don't get asymptotic freedom, we get quite easily asymptotic safety for such models Which is nice and the task is to deal with more and more complicated models where the complication are the details of the interactions and of the Well The sort of ingredients that go in the definition of the field to match a proper discrete geometric interpretation Okay, so at the non perturbative level, which is what is needed to To do the continuum limit to study the continuum limit There have been two main Directions, I will only mention the this I will only discuss a little bit more this first one based on the function renormalization group analysis, but there's been a lot of work on just constructive renormalization by more mathematical inclined people than I am so Let me just mention the result. It has been first Introduced this strategy by Astrid and Tim but in In a slightly different manner, but the we followed from that by Applying the functional renormalization group Alavetic mainly To again more and more complicated models starting from a billion ones with almost no geometric Ingredient and no extra complication motivated by discrete gravity Then we started adding all these little complications then we went on a billion. I Just highlight what what we seem to be finding in rather General terms, which is we we seem to find up to now in very simple truncations the one that we were able to study analytically Because most of the work has been analytic We generically find two fixed points a Goshen one in the UV and something like a Wilson Fisher fixed point in the IR and So we confirm the perturbative analysis that was actually rigorous. So it's more the other way around About asymptotic freedom in the UV and we find hints of a symmetric phase and condensate phase The condensate is interesting for what I'm going to say in the last five minutes Which is we look in particular condensate states of these field theories to extract effective continuum physics and cosmology, that's why this More formal work of renormalization group analysis is also directly interesting for the effective physics The first work on this a functional realization group on Groovy theories was in this paper called with Dario who is also here and Joseph, Benjamin Here I wanted to say something more in detail, but I mean I'm talking to experts So this is really a few slides that will really be for non-experts because it's really setting up the usual functional realization group analysis keep in mind that the scale is given by group representations and That the interactions that you had to add are of this non-local type So a truncation is not only a truncation in the order of the polynomial in the field But you also want to make sure that you include the all possible combinatorial structures at the same order and Extending the truncation sometimes means maintaining the order, but adding more of these combinatorial structures So already that is quite non-trivial. So the rest is the standard stuff, you know better than I do Okay, so the result is summarized by the previous slide. So there's not much more to say and this is for example at order four In a model of three dimensions meaning three arguments for the field The three possible ways in which you can pair the arguments So these are three possible interaction terms all at order four Okay, I skip the details and I use my last five minutes to tell you something about cosmology So so that you guess you can see some physics at least towards the end All right, let me let me skip this again This was an example in which not only we added the non-trivial combinatorial structure, but the Domain was non-compact. So we had to deal also with the IR divergences and take a properly defined Thermodynamic limit and the definitional thermodynamic limit could not be the usual one as in local field cures You have to think a bit harder also about this more basic Ingredients again, however, we find a Gaussian Ultraviolet fixed point and a Wilson Fisher fixed point and the same hints for a condensate phase in the IR Okay, let me skip this was another slightly more complicated model Okay, the task If you want to extract the continuum physics is to identify the relevant phase where you have a good Continuum geometric Interpretation for your states and dynamics and this is absolutely non-trivial because as I try to emphasize Generic states do not allow any Interpretation in terms of a standard geometries let alone smooth Okay, neither classical nor quantum is really different set of degrees of freedom in particular you want to match you want to Bridge the huge gap between this microscopic set of degrees of freedom and effective models for cosmology and The closest one to what to the type of formulas we're using would be something like loop on to cosmology because of type of variables we're using so I Here I don't want to go again into detail, but the the general point is that if you really think what The cosmological principle and in general a cosmological approximation is from the point of view of quantum gravity the idea suggests itself that Cosmology has to be understood as a sort of hydrodynamics of this Many microscopic degrees of freedom. So the task is really to extract the hydrodynamics once they gave you the atomic field theory Unfortunately, we have much less control than we have for systems on space time About the general symmetries Because if we knew very well the symmetries that the microscopic models satisfy Well, I wrote the hydrodynamics is the dynamics of the global quantities of the concert quantities. So they would not be much room So in general this step from micro physics to cosmology will involve a huge Cosgrain in both of the states and of the dynamics and this is a difficult try going to a condensed matter theories and Say, okay, this is the theory for the atoms now. Tell me what is the corresponding macroscopic? Physics from first principles If you survive the encounter, you will at least learn the lesson that is difficult And it's not something you ask so naively But we already heard this morning a thing that there is one example in which the micro physics dictates a little bit more what the effective macro physics should be those are superfluid or in other words condensates so that's a very special case and Because of that, that's the first one we study But we have a good geometric reason to To look at that. In fact, one can show that problem number one Which is to identify a class of quantum states in the fundamental theory with some continuum space-time interpretation Is well tackled by condensates of the group field theory formalism In fact, one can show and I'll skip the I'll skip the details Although I didn't receive any any bad look up to now from the chairman. So, okay, I'll go fast I'll try to finish and then in case I'll come back Okay, they would be I hope there will be questions anyway, so I don't want to take all my time so one what one can show is that Simple quantum group field theory condensates have the interpretation of continuum Homogeneous spaces so they're well suited for cosmology This is the simplest example Remember, we have a folk space underlying our theory. So we can create Particles acting with a creation operator, which is just a field operator You can even create an infinite number of particles by acting with an infinite number of field operators If you do it with this particular type of exponential operator, you do something special you create an infinite Superposition of zero one particle states two particle states three particle states all with the same wave function Sigma Which is the order parameter is the collective parameter for this all infinity or degrees of freedom You have a single function on a domain, which you can show If you've done things correctly to be isomorphic to mini super space So the space of homogeneous continuum spatial geometries This means the following that okay, I'll skip another bunch of details I'm happy to Come back to this is where I was It means that When you extract the effective macroscopic dynamics for such Condensate states what you're going to get is some generically non-linear equation For such collective wave function If you know about the superfluids and from the microscopic point of view, that's exactly what happens and in fact just following the usual procedures what we obtain is that the group filteri analog of the gross pitaeski hydrodynamics So the simplest form of hydrodynamics for a superfluid neglecting fluctuations and quasi particles and all these interesting features Is a non-linear extension of quantum cosmology Where the dynamical variable is the collective wave function that I introduced in the previous slide that governs this Infinity of microscopic degrees of freedom for those very simple states and which is as I emphasized a function on mini super space So at least is consistent, but it's hydrodynamics. So the equation you get is non-linear It cannot really be interpreted as the quantization of mini super space Okay, so we get an example in which cosmology comes out as quantum gravity hydrodynamics Okay, so this and the effective equation is nothing else that the classical Equation of motion for the group filteri field just like in BC now. I show you just one More concrete modeling one minute two minutes Okay, so we start from the favored well by many Microscopic model at the spin-form loop on to gravity level we write it as a group filteri This is called the EPRL model for four-dimensional Lorentzian gravity. The group is SU2 and The dynamics of the action of the group filteri and cause a bunch of extra complications needed to ensure the geometry interpretation Which I spare you We couple a free massless scalar field Plus we do a bunch of other approximations and the coupling is just an extension of the domain of the group filteri field From the arguments that only encode the discrete geometry to an extra real variable Which has to become the continuum free massless scalar field in the appropriate regime Okay, then we have to modify the model by adding appropriate dynamics for such extra variables Well, again, let me emphasize we do that by only having under control and not fully the discrete level We know that we are adding variables and putting conditions on this on these variables So that there is a good interpretation as a discretized the scalar field on a lattice on where we have also discrete gravity There's no yet any input from continuum physics Then we reduce to isotropic Configurations and this basically means that our collective weight function is a function of just one spin and the real variable representing the scalar field Okay, here Then we extract an equation, which is just the condensate hydrodynamics from the microscopic theory following the steps I mentioned in the previous slides Is an is a non-linear equation. You see this sigma to the fourth term For the sigma function a function of j one spin And phi the real field and these functions a j and b j come from the microscopic theory Okay Then we find that there are two concert quantities and we can also rewrite in more hydrodynamic like form the wave function the Madelunga representation And this mj just the ratio of the microscopic coefficients Okay, then we look at observables. We want to rewrite this Dynamics in terms of the dynamics for the total volume of the universe as defined by the microscopic states So we write the total volume knowing the discrete geometry interpretation of each building block Each building block is a tetrahedron. So that there is a definition for the discrete volume of each tetrahedron And you take the corresponding total second quantized operator and you compute it in the condensate And that's it and then we have other observables that correspond to the scalar field Which are consistent with the interpretation of having coupled a free massless scalar field So we rewrite now this equation as an equation for the volume as a function of the scalar field used as a clock Okay, in fully relational terms, there's no no coordinates no manifold here only the observables Well you obtain an equation like with two equations and you can show first of all that the volume remains positive on all solutions Which means that generically you have a bounds You never get zero Second you show that when you take a classical approximation you get the Friedman equations So you can really interpret this generalized equations as generalized Friedman equations With the quantum corrections giving you a quantum bounds and other people in london have shown that there is a corresponding acceleration Then we have shown that in some for some special choice of condensates you reproduce exactly the loop quantum cosmology dynamics And you can show that depending on the solutions you may get An acceleration that is lasting enough E-folds to avoid inflation to avoid the need for inflation This is all preliminary, but at least it's encouraging. Thank you