 Thank you very much. It's great to be back in Trieste actually. I realized I haven't been here for 11 years and just seeing Eddie Gava reminded me of the good old days in the 90s when I came here basically every year because I was one of the organizers of the spring school and have very fond memories of that and I also want to thank the organizers of the Exacter G conference for an invitation. It gives me a chance to meet some of the people whose papers I've been reading. Okay so the title of my talk is this and just a couple years ago I started wondering about UV fix points and I was actually led to it through some adventures in so-called higher spin ADS CFT correspondence which I'll mention very briefly and then actually with Simone Giombi who is an assistant professor at Princeton and a couple of excellent graduate students we've done a whole bunch of calculations. Some of them are not entirely original but I'll share them with you anyway. So this is the incomplete set of various papers and in the first part of the talk I mainly talk about the gross Neveau conformal field theory and then in the second part of the talk I'll be a bit more adventurous and talk about the ON model above four dimensions which is a somewhat less clear topic. Okay so the gross Neveau model it crops up a lot in particular in this conference in two so gross Neveau's original paper was in two dimensions and the idea was that it it's a kind of nice toy model for four-dimensional QCD in the sense that this coupling in two dimensions is asymptotically free. Just to explain notation I'll mainly work with four component the rock fermions for reasons that you'll see in a second so and it's asymptotically free but it's it doesn't give a CFT in the infrared because there is a dynamical mass generation. Now there is the physics is somewhat similar to the two dimensional ON nonlinear sigma model with n bigger than 2 and now we're interested in going slightly above two dimensions where both the ON and the gross Neveau models have weakly coupled UV fix points and then they essentially become conformal field series. Okay so just to remind you the beta function for the for the ON model there is this negative term this is for the gross Neveau model. There is a term which is negative for n bigger than 2 and n is for an F in our notation it's also the number of two two component Majorana fermions which is the minimal number of fermions so our n just basically measures the number of fermionic degrees of freedom. When this number is bigger than 2 you have a asymptotic freedom in 2D but then slightly above 2D you have a perturbative UV fix point and this was appreciated I think soon after the gross Neveau original paper and a bunch of people long ago developed two plus excellent expansions for various operator dimensions and then here are the answers you actually notice one interesting thing is that all the perturbative corrections vanish for n equal one this is just the bare dimension of psi and these are vanishing perturbative corrections and this is the bare dimension of sigma plus perturbative corrections so sigma is just psi bar psi so for n equal one something special happens to this model and I'll come back to this somewhere a bit later now there are similar expansions in the n sigma model with n bigger than 2 which is also asymptotically free but that model is a bit more complicated because there are infrared divergences in due to the scalar fields. Now a 4 minus epsilon so the upper critical dimension as you push above 2 the upper critical dimension is 4 and of course the ON sigma model is in the same universality class as the Wilson Fisher ON model which is just the 5-4 theory with ON and variant interaction and at 4 minus epsilon dimensions it has a weakly coupled Wilson Fisher now infrared fixed point and so using the two epsilon expansions the 4 minus epsilon and 2 plus epsilon one can actually derive excellent approximations to the critical exponents in D equal 3 they're not as excellent as the bootstrap values that Slava told us about yesterday but but they're nevertheless played a very important historical role and for example in Kleinert and Schulte-Frelinde's book it's done up to five loops and you get very good values and the field is still not standing still there is now a 6th loop apparently available so it's been a great way to approximate things even though the epsilon there is summation of epsilon expansion is of course a bit of a it's not an exact science but it still helps a lot now the gross Neville model so the question is what is the UV completion of the gross Neville model namely how do you flow to it from above rather than from below and the answer actually it's something I didn't know till I started working on related things and kind of guess the answer but it turned out that it was in the literature since 91 in fact Zenzhou's 10 and independently Hassen-Fraenz Hassen-Fraenzians and Kuti and Shen basically realized that it's just the Yuccava multi-flavor Yuccava model and now people often call it the gross Neville Yuccava model so you basically it has the same symmetries it has a it's renormalizable in four dimensions these are the two renormalizable couplings in four dimensions but now the difference is that as you go slightly before slightly below four dimensions these become now slightly irrelevant as opposed to slightly irrelevant so this model has it's a little bit harder to calculate with because there are now two couplings so you need two beta functions for two couplings and so these are the beta functions up to two loop order these were actually done in in a paper by Karkine and collaborators in 93 and we rederived them and rechecked them and so you see some interesting things like the appearance of the square roots and the fixed point values of the coupling and these are the couplings up to the order that we need and then you can derive the operator scaling dimensions so now sigma basically stands for psi bar psi in the gross Neville model you get this number then you get psi dimension and also another interesting dimension is the sigma squared operator so using the two again using the two expansions the four minus epsilon and two plus epsilon you can actually construct good values between two and four and now they actually two plus epsilon expansion is under a better control because they're annoying for a divergence is so so here is there is one thing you carry from the stock many of you know this already is this picture that so we have this interacting CFT which in D equals 3 is basically strongly coupled and non-perturbative it's wedged between two free CFT's you can either float it from so the directions of the flow lines are like this this is from UV to IR right this is so you can either start with the the usual gross Neville model is sort of this part so from the point of view of N3 fermions you have to add an irrelevant interaction and you get here but there is also a way to flow down to this model from the free CFT of N3 fermions plus one scalar and the nice thing is that so as you very dimension when D gets close to four this fixed point basically slides near here and you can do perturbations here in here here when you when D becomes close to two you can do perturbations here in here and of course D equals 3 is the desirable case but you have really very good control from from epsilon these two epsilon expansions and also one over N expansion is another a good thing about which there are many things known and here is an example of another quantity that we recently calculated and this is a completely new calculation it's the so-called CT the coefficient of the two-point function of two stress energy tensors it's determined by conformal invariance and here is the result of our sort of resummed epsilon expansions as a function of D it's a pretty interesting result because you see that actually so this is N times CT over CT3 minus one so when it dips below below zero that means that CT actually increases when you flow from the UV to IR and some people worry about whether CT always decreases under RG flow this is actually strangely enough violated very near very near two dimensions but in three dimensions it's not violated so it actually does decrease under the usual RG but then there is another measure of degrees of freedom which is essentially what Calabrese just talked about this morning it's related to a particular kind of entanglement entropy in two plus one dimensions of course he mainly talked about one dimension where there is no shape dependence but when you start looking in two spatial and one time dimension then you are on the plane and the simplest type of region is well people often talk about just the half plane entanglement entropy but more informative in a way is the circle namely you draw a circle of radius R and you compute the entanglement entropy between inside and outside of the circle and then there is the aerial law here is just a perimeter law that's a non-universal term and then there is a sub leading universal term which you often see denoted as gamma in the gap system it's so-called topological entanglement entropy but we started calling it F and for reasons that essentially in this case there is an amazing trick of calculating this so for this particular shape and in a conformal field theory you don't need the replica trick you can just map it using you can essentially map this region to the sitter space and think about it as a the sitter entropy and Euclidean the sitter is just a three-sphere so so this F is just minus the log of a partition function of the Euclidean CFT on a three-sphere so if you will it's a kind of trick that allows you to compute the entanglement entropy in a very simple way and so there is something that actually in the paper with the Jaffer is Puffer and Savdi we call the F theorem namely that when you flow from the UV to IR the F UV is bigger than FIR so this is what's called the F theorem in the three dimensions and it's believed to be a kind of analog of the zomologic of C theorem in two dimensions and the Carti A theorem in four dimensions and to flesh out this fact actually John B and I a couple of years ago considered this quantity in continuous dimension like in the spirit of how we consider the CFT in continuous dimension we can do the same on this for this sphere free energy then it's convenient to define this quantity sine pi d over 2 times log z the reason you need this something like this is because F actually blows up near even dimensions due to wild anomaly so for example in 4 minus epsilon dimensions you see like a 1 over epsilon pole and same in two dimensions it's really due to the fact that there is then a log R term there is a logarithmic term in radius of this region so when you kill this off you you find that the result is a is a completely smooth function of dimension so this F tilde thing looks like this so here is a result for free theory so what are the simplest CFTs in the world they're just a conformal a coupled scalar field or a massless fermion and these are the you have to trust me on this it takes a little while to derive these integral representations but you can recognize some familiar numbers for wild anomaly so for example this F tilde in even dimensions is just pi over 2 times the anomaly value and so you you see for example like you know for a scalar it's 190 and you indeed see pi over 180 the blue line is a scalar and the red line is the fermion and that's a completely smooth function of dimension it doesn't exhibit any features it seems to monotonically drop off as a function of dimension so now that that's the easy thing to compute just for free fields but then we did perturbation theory in in the interaction so for example for the Wilson Fisher model you just perturb by 5 to the 4 interactions and and you have to do integrals over the d-dimensional sphere this actually gets very hairy and but now there is software that allows you to do these integrals the really hairy part is that there are also in addition to the usual renormalization of coupling constant you also have to add one of our epsilon poles multiplying the Euler number like there is the so-called curve because you're in curved space so you have to you have divergences proportional to curvature squared terms anyways after many months of computation you get the following 4 minus epsilon expansion for the ON Wilson Fisher model actually up to epsilon to the fifth and and it's actually is amazingly informative when you look at the numbers you see that the leading interaction corrections are very small and and you essentially find that you find that for example for the 3d easing model at the interacting fixed point this f is only 2 percent lower than the than the free scalar so the easing model is just the 5-4 theory so you flow from a free scalar to this model I used to worry a lot like when we wrote this paper about that theorem over five years ago I worried how do you test this and for example what if like the easing model results overshoots and becomes negative which would be really bad too it turns out that it's extremely close to free field value and this is not too unusual because even the dimension of the five field that in at the easing fixed point is only like 3.6 percent away from free field value but this is even closer okay so so now we did the same thing for the gross Nebusse of T there is a four in this case there is nice four minus epsilon expansion and there is a nice two plus epsilon expansion so it's exactly the kind of thing that's the easiest to put a you basically construct polynomials that match onto the information both near 2d and 4d and these are the plots as a function of D for this f tilde for the f tilde minus n of the free fermion and you see that it's always positive so so this f theorem is satisfied not just in 3d but for all dimensions and it also matches onto a large n computation which is easy to do which I will not describe here okay so here is a summary of the results on 3d gross Nebusse of T's as a function of this number n of the fermionic components this is not entirely new I mean there are other related papers but I hope we did a more complete job actually there was a paper about n equal 8 just the other day by John Gracie and and collaborators who were actually a bit often one of their estimates but but so you these are the plots as a function of n this is n equal 2 3 4 and so on the values in this table you see that that large n everything matches on nicely to the large n results and now let me so so basically we've succeeded using this sort of traditional Rg just resumption of different perturbative limits to get some hopefully good information about this CFT and of course this can be compared with the exact Rg results and can be compared with conformal bootstrap results which I think the work is underway by conformal bootstrap there already was an interesting paper but I think there is more work underway and now let me talk about the special thing n equal 1 which is as I mentioned it's a very strange limit from the point of view of the gross Neveau model because if you take just the two component by Rana sidebar psi square it just vanishes so there is no way to write this model as a for formula theory but there is a way to write it as a Yukawa model so in this case the one of the formulation is just trivializes but there is still a Yukawa formulation okay and in this case actually it's a simple model for something people call emergent supersymmetry it's just an example of how in the IR there can be enhanced symmetry so this minimal 3d Yukawa theory for one two component my Rana and one to the scalar field was conjectured to have emergent supersymmetry I believe for the first time by Scott Thomas in an unpublished seminar in 2005 at KITP or maybe 2006 but somewhere around then but he never wrote the paper actually his slides are quite informative but I found them easier to read than some of the subsequent papers but but then so the amazing thing about this model is that the fermion mass is forbidden by the time reversal symmetry so all that you need to do to reach the CFT is a single tuning of the sigma field mass and then it's alleged that you're automatically getting a supersymmetric model so this could be an example of how something supersymmetry can show up not at LHC but in some sort of condensed matter system and indeed there have been some attempts to at a condensed matter realization at the boundary of a topological insulator so essentially you can take the UV Lagrangian to be this minimal kind of was a mean oh model in three dimensions for the n equal 1 super field and this is the bootstrap paper but they actually didn't have a particularly robust indication of this fixed point although once you look at it it's sort of an agreement with what I'm about to show you so now so what happens in this n equal 1 case then for as I said you cannot use two plus epsilon expansion at all it just trivializes all corrections is zero that just means that this models in two dimensions it's not a weekly interacting theory it's actually I'll show you what theory it is it's a well-known minimal model it's a strong still stays strongly interacting even in two dimensions but it is weekly interacting in four minus epsilon dimensions and one sign of things happening for good things happening for n equal 1 is that the square root actually equals 13 it's not some irrational number which turns out to be a lucky 13 and then you plug in this 13 and you see that the two couplings are actually related by a factor of 3 which is a sign of this what you see in the was a mean a model so it's consistent with emergent Susie relation and perhaps even more strikingly all the different looking dimensions start just differing by one half up to two loop order so this is in Scott Thomas paper he had like this part and it actually wasn't clear how he was even doing the form I you have to realize that this n equal 1 limit is a little bit formal because if you start in four dimensions with a myerana you have to kind of take half of a myerana but this trick works very well I mean it wasn't emphasized by anyone writing these papers before but we kind of once we started doing this we realized that this is what needs to be done and strangely enough it works perfectly you see these one loop and two loop corrections lining up so consistent with the Susie relation and it's nice that to see it order by order in epsilon expansion so of course it's tempting to conjecture that this is exactly true everywhere below four dimensions and it would be nice to test at higher orders in epsilon I'm sure and there are some expert experts who can do you cover theory at three loops and maybe four loops but we haven't done it and but we're fairly sure that these miracles will continue to hold now per day to D equal 3 gives delta sigma which is this value and it seems close to the bootstrap result properly interpreted but I think there is more bootstrap work on the thing is that when you do bootstrap you have to assume something about the dimension of the next operator and and where you are depends on what you assume about that we actually know what the dimension of next operator is from our epsilon expansion and if you look at that point in the curve it seems to agree pretty well but another amusing thing is this continuation to D equal 2 it gives an interacting super-conformal theory and it's actually just a tri-critical easing model which has a central charge now one of the beautiful things about this F quantity is that when you continue it to D equal 2 it literally becomes the central charge up to some factor of pi over 2 so we can using the extrapolation of F that they this quantity the sphere free energy we can figure out what the central charge of the model is and the epsilon the epsilon expansion just gives us point to 1 7 which is amazingly close to 1 5th which is the exact value sorry sorry this is for the for the dimension of sigma and for f over f scalar it's even closer it's 0.68 while C is 0.7 so one can track these operators from four all the way to two dimensions and and it looks like there is a set of models with emergence of persimetry everywhere in between okay other maybe any questions about this or yes I think they're pretty I think they're pretty close yeah yeah I there were yeah I may not know all the literature well enough but there were in particular estimates for unequal 8 for example in the paper by Harbutt and Janssen and we actually just recently checked that they're close to that although they they were also part of what they were doing is also like this G and Y model the gross Navier-Cawa and some use of that but yeah yeah I mean it would be really fun to compare I mean we have this table with the literature that we have found I think these tend to be in good agreement this unequal 8 is a rather special value which many people studied because it's supposed to be realizable and condensed matter okay so now so when N is large this is that's actually how I got into this stuff in the first place so Polio Kavanai conjectured in 2002 that essentially the special thing about these large N or N and G N model is that they have an infinite number of approximately conserved higher spin currents in the sense that in free fields they are exactly conserved and the flow makes them still does not destroy this conservation and it means that all the anomalous dimensions of these currents are of order 1 over N and that's just what you need to build an interacting sort of an ADS dual and D plus 1 dimensions so so there is something called Vasiliev theory in ADS 4 I should say it's a different Vasiliev from the one who did the the large N calculations in that model so so this is Misha Vasiliev and he wrote some beautiful papers in the early 90s on just higher spin gravity found some formulation in ADS 4 and by now there is a set of checks of this duality so so this kind of gives another motivation so by studying these models you're supposedly learning something about quantum gravity in ADS space but it's a rather special theory of quantum gravity which which contains all these higher spin fields in addition to spin to gravity field now so the ADS CFT relation for the dimension of the scalar operator versus the mass of the field is just this it's a very simple relation and you can have both basically both the delta plus and delta minus values for some negative values of M squared I should say that negative M squared in ADS space is not necessarily unstable it just has to be above the bright and loner Friedman bound this is a very famous result in ADS space the bright and loner Friedman bound and in this range you can take both of these dimensions and actually if you look in ADS 4 say for the gross Neville model there are two values delta equal one and delta equal two and one of them corresponds to interacting theory and the other one to free theory so so this is and the same thing happens for this for this interacting oil model so it's a very promising thing to start comparing the 1 over n expansions so now I'm going to make a transition to something a bit more speculative like the the gross Neville model is like it definitely makes sense I think and it makes sense for all values of n and hopefully there will be more experiments yeah we've also done work on QED 3 which there is still a question whether it's a CFT for very low values of n but but there is a similar story at large values of n but then so you build the 1 over n expansion by using the Hubbard-Stratanovich auxiliary field for example in the ON model and this you ignore at the fixed point that's the sort of old method of which I don't know who applied it first maybe Ken Wilson but then it was really pushed forward by Vasiliev and collaborators and in the Leningrad group and so so you get like this the nice thing is that you get a non-local effective action for the Sigma field and there is a kind of induced dynamics you see the this non-local propagator emerging and then when you transform it back to position space you see that the Sigma field has dimension to up to 1 over n corrections which is the right value of dimension and then you can build the 1 over n expansion for the dimension for example of the Phi field just to make it clear now I'm talking about the ON model I'm no longer talking about the Gross Neville I'm talking about ON model so the dimension of the Phi field will have the free value plus 1 over n 1 over n squared corrections and people persevere to compute these functions as a function of D like so you know for example this eta 1 here is its explicit form as a function of D so it's a kind of non-perturbative thing where it it requires a certain diagram resummation and it's actually known up to order 1 over n cubed where it starts the formula get like half a page long and they involve all sorts of special functions and in fact some of these old papers on top of everything had typos and so it's a bit of a take some work to get into this field it took us quite a while to sort this out but one of their big tests was that you can compare with for my for minus epsilon expansion sorry okay so so here is something that is evident in this result even for the leading 1 over n so if you plot this as a function of D this is what you see it's positive between 2 and 4 and if it once it goes negative you clearly see that the large n series non-unitary because because this this is just the unitarity bound right so if this is becomes negative then you know for sure that a theory at very large n is already non-unitary but luckily it's clearly positive here and of course we all know that Wilson Fisher series unitary and nice between 2 and 4 but then it's again positive between 4 and 6 and then above 6 it really goes non-unitary and the two-point function coefficient C is has similar properties and we actually for reasons of this studying higher spin theory we wanted the 5d theory to be sort of okay like to still be at least unitary at large n and this is actually how we started worrying about the five-dimensional n model so so we started thinking could there be actually some kind of ON model formulation between four and six dimensions which is at least at large n unitary and at first it seems a little crazy because like you open Michael Paskin's quantum field theory book and he immediately tells you don't look for scalar theories above four dimensions they're all trivial right so you would need some kind of a UV fix point of this model but then when you start doing actually I didn't find it myself but Parisi told me that he wrote the paper in 75 on non-renormalizable interactions and literally discuss this model and this is sort of an old topic like if you go to four plus epsilon dimensions you see that there is a UV fixed point but it's at negative coupling and negative coupling it seems weird but it's not deadly at large n right you you can have a theory sort of sitting at the top of the potential at large n and still be sort of okay so our goal was mainly to study the theory in five dimensions because there is some indication of the existence of of higher spin theory in ADS6 which would be due all to this five-dimensional theory and then we started sort of thinking about what could be the UV completion of this this non-renormalizable theory and arrived at this and this we did independently for what it's worth I mean I don't think it's very profound but we we haven't found literally papers on this so it's surprisingly so it's just a version of our own symmetric cubic scalar theory and by quite analogously to the gross Navier-Cawa model you have to add an additional scalar field and write a model with sigma phi i phi i term and sigma cube term actually we realized this before we even knew about the gross Navier-Cawa model but and then we started playing with the beta functions so this was about two and a half years ago or more we started playing with the beta functions and we saw that there is an ir stable fixed point there is a real solution in six minus epsilon dimensions provided n is very large so so if n is above 1038 there is a solution if n is below 1038 it goes complex which is a pretty so if you draw the two like there are two curves that intersect when n is bigger than 1038 and and so this is the IR fixed point of this cubic theory but then the intersection disappears and that's a pretty typical situation that you can encounter like the there are two fixed points that merge when n hits this critical value and then they go off to a complex plane and they but it's sort of nice that for large n everything matches with the large n philosophy because there we could use just this large n philosophy and if you look at the so then you can do all sorts of perturbative checks with large n and they all agree and I should say that the the model is quite different from the original five cube model in six minus epsilon that Michael Fisher studied in 78 he studied the n equals zero version in that case the fixed point set at an imaginary value and that was manifestly non-unitary but stable manifestly non-unitary but it's actually stable because the past integral just oscillates and he did it he managed to estimate the liang edge singularity by continuing from six to two dimensions and this was before the exact solution of beloved Palakov's homologic was known so so this this was a kind of historic paper and but then we realized that there is a totally different sort of thing where n is large and the solutions are real and then we did the three-loop analysis the beta functions get complicated but then the nice thing is that we can match the six minus epsilon expansions of because you see you you know these in arbitrary dimension from the work of of a serial at all and you and you see that our six minus epsilon perturbative expansion exactly reproduces these so if you will it kind of gives you a new cross check on both the large n formula and the fact that the series is like the uv completion of the online model and then John Gracie did it even at for loop and it continues to work so so it's pretty non-trivial checks because you can compare many many coefficients here you can go like so the large n formula work to all orders in epsilon our formula work to order epsilon cube but to all orders in one of our and then you can compare the between some set of the two coefficients but then the question is could there be what happens in 5d and that of course gets non-perturbative and then one can try to do exacter g or some other approaches or one can try to resound the epsilon expansion and if one just naively proceeds by computing one of epsilon corrections you do see that 1038 gets corrected downwards twice pretty sizably by the two epsilon correction so if you just plug in epsilon equal one you you get down to a fairly low value but then John Gracie's value I didn't show it here is positive and largest so it's basically hard to tell but there is a kind of elephant hiding in this closet which is does it even make sense to talk about critical n for this model because it's just a cubic model right so so the path integral is not maybe strictly well defined so does the theory makes sense non-perturbatively so we actually did some work on this but never so far haven't written it up but we believe that at least the vacuum at at 5 equals here and sigma equals here is metastable because you can use standard instanton methods to compute tunneling like when you're in six minus epsilon dimensions you can just do perturbative Rg and the potential should be close to to the bare potential and the potential just looks like this right it's just looks like this so you to compute tunneling you need an instanton that goes from here to here and it's not hard to get this exact solution in fact it's in a paper by Alan McCain from long ago but we had to adapt it to this large n setting and essentially you find that the model is at least metastable with a long lifetime when n is large and epsilon is small so the danger is that you can pick up imaginary parts due to this instanton of order e to the minus n over epsilon but we haven't fully finished this project so we don't know for sure that these imaginary parts really spoil the game and then there was I guess we will hear in the next talk about some of the work from exact Rg which also suggests some possibly metastable solution and then there was some work from conformal bootstrap directly in 5d and it's fairly recent paper and it actually looks fairly encouraging for the existence of some model and five dimensions so they look that n equal 500 which is about half the value the critical value in 6d and they get these islands the type of islands that Slava reviewed in his talk that are definitely there and decree and shrinking for the equal 30 n model so you get similar types of islands and the large n result which is the only thing you can easily do for this model sits right in the middle of this island so so it looks like this these islands are kind of shrinking around what where you think the scaling dimension should be according to the large n approximation it would be very interesting to pursue this further and see if this is if these islands keep shrinking and how precise this is okay so so so the so let me conclude so I've talked about some adventures in in what seems to be the method of yesteryear which is time honored method this was long before my time in the field but I was told that the Wilson Fisher paper was quite a breakthrough because it gave a kind of quantitative tool to to approximate series and it happened to work very well for the ON model granted it doesn't always work very well but but it appears to continue to work well for gross Neveau model our paper also considered the Nambuyon-Alesinio model which is similar to gross Neveau but it has an extra U1 symmetry and then there are other vectorial CFTs to which one can apply it so why is this useful well Slava quoted my friend Poliakov saying that RG is a human-made thing but I mean it's good it was made at some point so it's certainly you know it's hard to be no for sure outside perturbative region but and you can miss some theories but when it works well it really tells you where the theory is and then this is for example a sign of like so here the RG tells you that this is where the theory is and so in the cases so far where bootstrap was successful it does agree with RG at least in almost all the cases maybe bootstrap gives more precision so I think these studies of RG give you a way to pinpoint where the theories might be and facilitate the life of people doing bootstrap because they can tune things to to find these theories more precisely I also talked about the RG inequalities like C theorem I theorem and F theorem they all give you a quantity which if a CFT if a short distance CFT flows to a long distance CFT something is supposed to decrease now in two dimensions I'm allergic of proof then in 87 that I vaguely remember first it wasn't clear how much of a breakthrough it was but it kind of set the gold standard for for these theorems and then for a theorem there is a conjecture by Cardi from 88 and then a recent proof by Kamorgotsky Schwimmer for F theorem there were conjectures a few years ago by our group and by Rob Myers and then was proved by Cassini and Huerta and now what I showed to you is a kind of possible interpolation between all these theorems just by studying spheres and continuous dimension because all these theorems actually come from spheres and continuous dimension and you can interpolate from A to F using this these sphere calculations and and also like these these things are useful for ADS CFT were in addition to gravity under higher spin fields now some small values of N can be very special because they're enhanced infrared symmetries for example for Yukawa CFTs one can exhibit emergence of per symmetry in the gross Neville Yukawa model one sees it for minimal number one myrana fermion we also have results what they're also not fully ours but revisited by us a two loop order by where you take a complex scalar and and a Dirac fermion and then you have more control because there is a kind of you want our symmetry so you can see these emergent symmetries for low values of n and then more speculatively we found a possibility of our n models and deep between four and six which I believe are at least metastable and what does it mean no one is completely sure yet but I think one possible interpretation is that that there are small imaginary parts to the scaling dimension which means that there is no true second-order phase transition but the transition could be very weakly first order and if that's the case maybe that can be seen in actual simulations so so there have been other approaches to the 5d and model other than our attempts to do it via epsilon expansion namely conformal bootstrap and exact Rg and I'll just end with a question like could the phase transition and 5d be very weakly first order for large n I don't think anyone really simulated and very large for on magnets in five dimensions but I think if we are right that there is some kind of metastable theory maybe that's what happens but okay thank you for your attention