 Okay thanks Eric and thanks for asking me to this meeting. It's a pleasure to to speak at Derek's Celebratory Meeting. It's a double celebration because it's 25 years this year since we first met in Pisa in April at the ACAT meeting where David and Derek were exponentially developing the knot theory and subsequently wrote a paper together and some of the work we touched on will be mentioning later on but others have sort of quite correctly given an appraisal of Derek's support through their careers and I've certainly benefited from that through this Mercator Fellowship that Derek got me several years ago but that's just another aspect of Derek's support for the community at large. He also just organizes our meetings for us and gets us together and gets trained it together and gets us working together and I think we should be very grateful for that and it gets us to meetings like this where we share different techniques to develop the subject in particular QFT but I'm going to speak today on a piece of work that came out of work with colleagues in Germany, collaborators in Germany, Andreas Tynand Cologne. We're working in CNES matter theories and this works based on an older paper and more recent work that's been written up at the moment but since we finished that project I've suddenly realized I could have done a more general calculation and this is what this talks about which is where the non-Abelian symmetry comes in, excuse me. So what I'm interested in and what we're all interested in the structure of QFT and the development of the precision calculations to be able to understand the structural aspects and also get numbers out for real-world phenomena and that's mostly in this area tends to be particle physics, precision put out to beta functions to high loop order and trying to see beyond the standard model but there's other areas of science that QFT underlies and one of them is material science and condensed matter physics and that particularly typically the connection with continuum field theory is in phase transitions in materials and in the last 10-15 years with the development of this wonder material called graphing people are looking at these materials now to try and understand new phenomena. A graphing is a sheet of carbon atoms joined together in a hexagonal lattice it's one atom thick which doesn't always mean it's flat it can be slightly crumpled but it's got a rather curious property in that it's incredibly strong for something so thin but if you stretch it or contract it a bit it undergoes transitions from what you'd expect carbon to be able to be inserting to more conducting phase and if we can harness this as a material it opens up a lot of opportunities to do develop new materials and a microscopic scale. So what I'm going to focus on is some of the theories that underlie these phase transitions in the quantum field theory and they're not unrelated to models we have in particle physics and the standard model. The key thing I've been talking about today is this gross navoia-calvotype interaction which is basically a scalar field with permeants and no structure in terms of large structure however because we're working with electrons you can add in QED quantum electric dynamics to govern the dynamics of the electrons as they move around the sheet of carbon atoms. I'm not going to focus on that today we've done working that before but the other connection one of the other connections is that because the similarity is underlying models which are like four primary interactions in low dimensions with two and three dimensions we could get an idea of looking at spontaneous symmetry breaking in the standard model by the mimic of the theory that undergoes underlies the phase sensations and the more complicated standard model spontaneous symmetry breaking but in the scenario where you don't have to dig a tunnel underneath Geneva which is 25 kilometers round and it's a bomb of a lot of energy into it you can do it in the lab that's the hope whether it's completely in what a parallel is another question but offers an opportunity to look at phase sensations in the simplest scenarios but the main issues the phase sensations are not only related to the Wilson Fisher fixed points of quantum field theories that we look at the Wilson Fisher fixed points are something which we in fact use in dimensional regularization in effect when we look at the basic renormalization before we put the dimension to be back three four dimensions or two dimensions or whatever I've said the focus here will begin grossed of a model but the the insight I've gotten recent years working with the condensed matter committee is that they endure these theories now with what is I regard as an unabillion symmetry which for us who do gates theories is second hat it's it's basically another structure groups theoretical structure we've seen talks this week on double copies and other aspects of the group symmetry restrictions on the field theories that just becomes an extra layer which we can handle but the other interesting development in recent months is a new universality class that's come out of studies of spin models so on the condensed matter side people look at spin theories where they have a discrete lattice and they put poly matrices at the at the corners and they describe the the phase sensations with lattice type models and these models are also in the same universe and the classic criticality it allows you to look at phase sensations but one of those spin models has given rise to a generalized gritt above model which is called a fraction light one and it's got a different group structure and that's the focus that collaborators have been looking at which is where I was brought in to do some calculations so that's the model I'll be focusing on primarily today so let me just go back and just do a wee bit of basics on phase sensations I'm going to talk about continuum field theories in other words honest to goodness data mental field theories with no discretization of space time and normally we focus on calculating beta functions and those beta functions are to allow us to evolve a coupling constant to high energy scales and low energy scales in standard model or qcd quantum chromodynamics to be able to have some idea of what what the behavior observables are which are objects in nature which we can calculate on the theory which are independent of gates parameter independent of the renormalization scale but there are other what I call observables or renormalization group of variant quantities that are much simpler to derive and they come from phase sensations or fixed points in the renormalization group flow so fixed points are defined by those coupling constants which start site beta of g equals naught so this equation here if you can see my mouse the solutions to this define the fixed points well it's typically given a one coupling theory but you have a vector of beta functions with a set of coupling constant every beta function by analysis and set of points puts all the beta functions by this define the fixed points so it's true for multi coupling theories as well there's several types of fixed points the the trivial one at the origin is known as the Gaussian fixed point and it's the one which is where we typically do perturbation theory around uh coupling constants are smaller in the neighborhood of the of the origin and so we can have an abandoned perturbative approximation in that region and get back to the free field theory which is what the Gaussian fixed point is however the more interesting one is that is the one that derives from the Wilson fixed fixed point and this is defined by the d dimensional theory so if you imagine that if you could calculate in d dimensions or d not the critical dimension of the theory but the dimension slightly above or below or somewhere off into the complex plane away from the integer dimension that you're working in then the beta function develops a term at the start which is order epsilon for epsilon is close to the critical dimension of the field theory okay now this is basically reflecting the dimensionality of the coupling constant a and b are the one and two lip coefficients of the beta function and so on now I've typically written a and b here as uh epsilon independent and that means I'm in the ms bar scheme if you're in a scheme which was not ms bar in other words with a finite renormalization these coefficients would depend on d or the epsilon effect and that would be a different uh scheme uh but let me just what I'm saying today is it's independent of the scheme so it doesn't quite matter what scheme I'm in but the fixed point is defined by the zero is the beta function if I take the first two terms here obviously I can get the zero from g equals not but I also get a non trivial zero which is order epsilon from the origin now say epsilon is a complex variable so the coupling could be complex and we overlooked that in d dimensions because we're not in a reality situation so issues about unitarity breaking etc are not really relevant it's only in the limit going back to four or two whatever is relevant and obviously the fixed point depends on the coupling of the coefficient of beta function which can include group structure and all the symmetry properties of the theory and there's obviously corrections from the b term and so forth all the way down the line and since I got a deal with the growth of our model this is an example of the one loop term it's got n dependence where it's an su n symmetry I'm putting f on the group there because it's a flavor symmetry now what's the point of the fixed point well if it was a gaze theory the gaze dependent object but despite that you can calculate gaze independent scheme independent information from the fixed point and it comes through what we call critical exponents if I take the wave functioning on my station group function on the value of the at the fixed point I get something called aida which is the wave function and almost dimension I'm not going to be a pure number or going to depend on the group casters or any parameters in the theory but it's independent of scales and it's a physically measurable object of course we truncate a series to certain orders and perturbation theory and by that we sort of have to terminate and so it would be truncation errors errors but it's a measurable quality and that's what the phase transitions are governed by these exponents obviously you can calculate all the ones the critical slope is a correction to scaling exponent and obviously the beta function balances at the fixed point but the slope doesn't it's a renormalization right again this is measurable and these to find the properties of the theory in data mentions obviously I did the perturbate expansion up here with order epsilon but in principle these are d dependent and can exist beyond the critical dimension up to any dimension you like so they're very universal properties of the theory in all space times okay now how do we work with this well we have to do an approximation um there's various approximations multicolor those things numerically uh there's the dimension can form a field theory in more recent years functional renormalization group where you can evolve operators across the spectrum using the Watson renormalization group uh formalism uh you can do perturbation theory and then use maps to product approximates between various dimensions and get a behavior and then there's the large n expansion which is what I've worked on over the years and whatever you do you have to do something just something to uh get to the dimension of interest and then condensed matter theory is three dimensions which is always one away from two or four so when we do epsilon expanses we have to sum down from four or sum up from two and that's not straightforward as some of you are aware now there's a secondary aim in this talk uh is to look at aspects of the large n formula isn't for another problem I'm interested in I've decided to do this for a general lead group um the problem with that would become apparently putting in a non-novellian symmetry is trivial in some sense but the motivation is to try and understand how the higher order calcimeres that will come in and perturbation theory come in in the large n expansion and this is a little toy model where you can do this so that's the secondary issue because ultimately would like to be able to understand gates theories with a non-novellian symmetry and similar methodology so let me get down to Lagrangian's uh we know the four fermi interacts in four dimensions is normally normalizable but it's renormalizable in two and this is the gross nebulin Lagrangian that was written down in 1974 and it has got several very interesting properties it's renormalizable it has the same background to quartic theories with scalar fields in four dimensions but it's asymptotically free in two uh I've got an SUNF I've got an SUNF labor symmetry but as with any quartic interaction you always linearize the interaction uh by introducing auxiliary field and it's this Lagrangian which is the key thing to write all the calculus to do obviously you can put a mass and critically masses are irrelevant because they don't scale the masses run zero fixed parts of scale free so the masses are irrelevant the sigma field here doesn't propagate but it defines the dimension of the sigma field through the interaction. Now one interest for models like this is that they got popular again in the early 90s because they could be models of composite Higgs fields where the sigma would play a role of a so the sigma would be the Higgs field and then side by side would be the composite fields at the fields that make up the composite Higgs field except this is two dimensions you'd have to have a four dimensional theory well you can't construct a four dimensional theory with that property it's called the Gross-Nivouria-Cavan model and it's termed the old development completeness and what that means is this is not renormal this is not this is super normalizable forward and make it renormalizable the sigma field has to become an already kinetic term but if that becomes an already kinetic term you have to pre-quartic interaction and that would be your Higgs potential and this was shown by Zinjusen in the early 90s uh so that would be the corresponding theory of four dimensions that's in the university class of this theory in two because it's the same core cubic interaction in the same core uh fermionic kinetic term and this operator is not relevant in four the quadratic vertex would be relevant in four so the sigma field depends changes across dimensions. Now the extension that people looked at for the condensed matter problem for the Mott transition years ago was a variation in the Gross-Nivour model with the karel-heisenberg one and the only difference is to put a poly-spin matrix in here lambda on you can linearize it but you have to have a vector of scalar fields and I've written down the other value completeness of this theory uh it's just uh by analogy so the pi field would have to have an already kinetic term the propagate and obviously have to have a quadratic term for renormalizability but the same core interaction here spawns the other interactions as you go through the dimensions if you go down in dimensions this becomes irrelevant um this becomes irrelevant but the other operators become relevant and that makes renormalizability in two dimensions but in between two and four there's a three-dimensional theory that describes the pair sensations in the carbon sheet the graphene sheet. Now the fractionalized gross-nivour model that I mentioned earlier on the more recent one is a variation in that where you replace the polymer matrices by SO3 matrices and the SO3 matrices satisfy this relation which you can just encode into your files and that's where I came in on the more recent work with my colleagues is to do the SO3 calculation so I reworked all my SU2 calculations with a different group theory structure and then Dumbo realized afterwards I should admit it for an already lead group and start it from a more general case with the usual group casamers then it could have taken the SU2 and SO3 results as a corollary of more general result so it's a more general universality class with non-abelian symmetry that I want to focus on now and go through the construction and basically the large n-expansion technique which I use is to you work completely now in the universal theory with facility technique from the early 80s and just apply the formalism from scratch so I mentioned earlier on about the ultraviolet completeness well if you extract from what that, if you extract the lesson from that basically means the large n theory will work with this Lagrangian we've got the core kinetic term and the core interaction with the non-abelian surgery and some function of all the other scalar fields and that varies as you go through dimensions but it's irrelevant for the large n expansion I call these spectators and I've rescaled a coupling constant out here to have unit coupling because it criticality there is no coupling constant it's just replaced by some number so you just start with this version of the theory and this comes along for the ride everything in this f of pi in the fixed events is contained within this interaction as effective vertices so life becomes a bit simpler the other aspect of the formalism that we work completely in coordinate space and I want to focus a lot more in coordinate space and have in the past because Oliver is giving a talk tomorrow on his graphical functions and the similarities but differences and I want to sort of add some some of the ideas to that debate this week and so in coordinate space the propagators all have a scaling form with potentially corrections to scaling which I come back to later but the key focus is on the leading term and the dimensionality of the field in the Lagrangian at criticality determines the power of the propagator and if you look at the dimensional analysis of the Lagrangian these the Fermion field has dimension which is d over 2 so I'm setting d equals 2 mu here just for a shorthand and the the pi field has a propagator which has got 1 but there's corrections to both and these are the anomalous space the mu and the one are the canonical pieces and eta is the anomalous dimension of the fermion field as I had before and chi pi is the vertex anomalous dimension of the operator that we have in the theory as well so both operators have anomalous dimensions reflected by chi and eta and just for the just for the future the this combination of exponents is d plus 1 minus chi and chi and eta will have order one of n expansions starting at one of ren with mu dependent or d dependent coefficients which also depend on the the group casamers and the beauty of the facility of method is you just take these propagates and put them in swingard-eisen equations and we'll hear more about swingard-eisen equations later and you just self-consistently solve for the unknowns which are the exponents and a an amplitude for a combination of a and c a and c are just x independent objects just to get a scale that's a good number on normalization for the propagators and that's quotes all you do and these are the leading order diagrams the next leading order diagrams with the two point functions uh of course you have to calculate the integral but in principle this is all you do you look up and you note that in the diagrams are written down there's no dressings of the propagators and that's because the anomalous dimensions automatically incorporate the the the self-energy corrections on each line so you automatically sum up those diagrams by taking the generalized anomalous dimension what you haven't done is got rid of vertex correction so there's two diagrams at one of our n squared the vertex corrections uh these two which are actually divergent as you would sort of half expect in quantum field theory but you can analytically regularize with the delta and if you do that it's possible to calculate them and have simple poles in delta at this order and double poles in next order now how do we calculate we use what's known as conformal integration or uniqueness so the rule for uniqueness or conformal integration is that at each vertex you have a sum of exponents you integrate over the vertex point so this is not like for momentum space where you integrate over loops you integrate over the points where the edges join but the the powers of the propagator joining that point are not one that there would be a perturbation theory they're different i've taken generic alphabet in gamma here but the rule is this is not possible to integrate except in a restricted set of cases the countably infinite number of cases and this the first case where you can integrate this vertex is when alpha plus beta plus gamma is 2m plus 1 so it's d plus 1 if that condition is satisfied then you can basically calculate the integral you get some overall normalization factor where it's just products of gamma function which i'm not going to dwell on these always come along for the right every time i do an integration and you replace the vertex where you're integrating by just the product of three propagators this is sometimes called the star triangle relation but it's in coordinate space there's a parallel of it in momentum space of course um there's a duality with it if the line the propagator opposite the point for you the opposite there is some propagator you integrate over is dual to it so it's mu space time dimension of two minus the exponent if this fermions is adjusted for the the x slash otherwise slash just plus one now seven aspects need to point out for the ghost number you can with carol heisenberg growth text two alpha plus gamma is two mu plus one minus chi high so unfortunately the vertex is not quite correct for uniqueness in the large n set up however with the regularization and you can calculate with subtraction methods to be able to get around that with still using uniqueness so the ways around it and the second point is there's various ways of proving this formula one of which is just do the Feynman integral uh either yeah the Feynman integral associated with by using the condition and you end up with hyper geometric function which simplifies into a geometric function when the condition is satisfied and you get the answer on the right hand side that's brute force the other way to do it is by conformal transformation and we'll mention those later on and the rule for the conformal transformation is that when any line joins so you define an origin for a an external point and then take a conformal transformation so the rule is that your case every line joining the origin by two mu minus the sum of the exponents joining that point which here would be two or two mu plus one in the case of fermionic vertex it would two mu plus one minus alpha minus beta minus gamma which is not if the condition is satisfied so then you just get an integral of a bubble or a chain integral which is what this is effectively and that's the rule there's no rule for qcd because it's got a gamma matrix inside it um no just applying the formulas we get these results first order next deleting order and the reason I'm showing these is several points you see the group customers coming in from before cacf and also this combination of side functions coming I've written this side function so it's got a zeta expansion near two two dimensions and it's these are typical of what you get but the leading on next to leaving order is true able to get the correction to scaling that I had in the which had the exponent lambda can also be incorporated in the formalism and in the kinesis model language is the correlation scale exponent and it's slightly more involved because you have to calculate the two-point function scaling function before you solve the equations and involves a horrible set of gamma functions incorporated in this notation but there's a subtlety with the gross neville case which I want to point out to you now it's to do with this function q that comes in mu no that's our lambda at leading order is mu minus one dimension of space time over two minus one and if you look at that formula with a specific example of the exponents we've got it turns out this is singular and that singularity means there's a reordering of diagrams in the Dyson swing equations so you need higher order diagrams to be able to go to one over n squared to get the second order correction to the exponent lambda this means just more diagrams to calculate and these are the diagrams for the dot indicates the insertion of the correction to scaling so there's a vertex correction on the trulip self-energy likewise this one but the insertions in line there's a vertex correction on the external leg and then they get the usual characters the primitive norm planer on these other diagrams which are well norm planer as well with these various decorations with the insertion now i've illustrated these graphically because those were two gates series automatically go light by light which is not something out of star wars it's basically this set here if i cut down this line i get four external wiggly lines which you could write as photons and that would be two goes to two photons scattering and these are regarded as tougher diagrams but there's two bolted together here it's really there's two bolted together here and each one's got a a lead group generator at the vertex so there's a lot of groups theory associated with this which we have to take into account for the calculations and the typical combination is written here a and b are external external group indices and cd and e are the internals i've written them sort of different orders here because the way the the the pie field goes across varies from loop from the left loop to the light loop but it's proportional to delta a b no in the mid seven in mid 90s uh from astron from mid bergen and laren calculated the uh forwarded better from security and they had to take account of group theory graphs or graphs with this type of group theory and they have a very elegant package called color dot each which allows me to do the group theory for this problem and it involves rank four fully symmetric tensor d abcd the f here means the fundamental what could be an adjoint for example there's a rank three symmetric tensor which we all know about in league groups which is d abc and this calculation stops absent well i get the d abc uh tensor in uh in the problem which is uh going to be products of these two things when when there's a shake down on the calculation so it's possible to code this in automatically and run through the calculation and this is the second order correction which is sufficient it's significantly more complicated than the original one for the uh the originizing model first of all a case from mid 90s which especially if i did separately but you see the rank four customer is coming in here uh at second order large n and there's seven new functions here that come in this five comes in uh there's a lot of decorations and the rest what's on the next sheet unfortunately the formula got a bit big and there's this data here which are the uh second so the derivative of the of the side function and these incorporate status so if you were to expand these objects out and that's an expansion they are four dimensions they would correspond to renominase this group functions near four dimensions or equally two dimensions if you did it around two dimensions and this would show you that two orders in large n that correspond to renominase this group function to all orders at this order in large n would only have uh restaurants and status uh it's not a proof it's just an observation because obviously there's going to be non-status further down the line uh at higher order large n higher order productivity it's somewhere you'll get them but the key thing here is now i've got this general formula i can specify groups very simply and take limits i'll talk about that towards the end now the other aspect of the facility of work from his school honking in and pismac as well was that he pushed the calculations to one of grand cubes this is i can't stress how much this is a well ahead of its time uh this was in the early eighties um the formulas was set in stone earlier on by polio cough in 1970 on perazy uh a few years later where they were doing what's now known as the conformal bridge strap method in essence though i'm using what's now known as the large n conformal bridge strap method which is what i've had to rename it after conformal bridge strap people uh come in and corner that market but if you remember before i didn't have dressings on the two-point functions but i had vertex corrections this method goes around that by getting ready rid of vertex corrections in these self-consistence uh serendice and formalism so these are the diagrams that contribute to the calculation of eight or three uh you calculate the vertex functions and through a formalism that i'm not going to go through again the um eight or three right these are all primitive diagrams because there's no vertex corrections but the dot that's on each vertex is not accidental it's deliberate the dot represents what's called a poly cough conformal triangle now if you remember that the vertex of the theory is set in such a way that it's not unique but the poly of course conformal triangle uh is in some sense mimicking not the absence of uniqueness but allows you to calculate these diagrams exactly so let me explain what a conformal triangle is so if you must an ordinary vertex with arbitrary um exponents meeting on a point poly cough replaced that point by a triangle with arbitrary a one and a two a three exponents internally obviously there's some proportionality function here would you choose these internal exponents which are completely unseen and not relevant to the field for each vertex by making each vertex of the triangle unique so these are the conditions here so if the triangles are only neat then there's lots of tricks you can use to evaluate the integral and this is the conformal transformation that you use you say x goes to one over x but for a vector it's x me equals x me over x squared or if you take um what would happen in the product of in a propagator would be x slash minus x y slash goes to x slash x slash minus y slash y slides over x squared y squared with a minus sign but if you're both a lot of these together with a y slice minus z slice there'll be a y slice here and another y slice for the next propagator it's one of the y slices it's a pair of y slices cast off a y square so you retain the same string of gamma matrices around a diagram after conformal transformation as the original diagram clearly because there's no gamma matrix in the vertex and that allows a lot of tricks to be used so with these vertex were placed by a conformal triangle you can calculate the diagrams but leading order as david reminds us in many occasions um their hardest diagrams sometimes occur are the previous loop order to the higher order in the expansion parameter and that's the case for the leading diagram which is just a triangle so this is the full setup for the conformal triangle uh uh definition of the it's a one loop triangle diagram so each vertex is now replaced by its conformal triangle these internal ones are chosen purely because of the alpha here here and here and the external one i haven't drawn to the top there's a regularization coming from the external lines because the equations as i said before are divergent so you have to regularize each diagram and there's two regularizations the delta here and the delta prime here and i've arbitrarily put the origin at top so how do we evaluate this well every vertex now is unique if i take a conformal transformation i replace every line joining the origin by the two mu plus one minus the sum of the exponents at a vertex if you do that you get this it's the same diagram but the zero this line is removed because the sum of the exponents at that line are two mu plus one now that means the only points to integrate are the ones on the hexagon that goes around here x and y are external so this immediately gets you down to a two-point function and this is an integration point and that's an integration point so again immediately integrate these two diagrams and i'm not carrying factors through this in the in the slides but you imagine the state of paper beside you where you carry the factors and you get this diagram which is two-point function which you have four integrations on and a lot of problems but this is where you have to think and the facility of trick was to write this as a diagram that looks the same without the top line removed but you multiply by a function which represents the discrepancy or the compensation for the absence of it this can be evaluated because it's just again simple chain integrals and you get a result which is this sorry this which is you get a result but the the compensation function can be written in this form or basically it's because it's a function of this combination of parameters so the orders were calculated and then you can write down the expansion it's in an exponential because it's easier formulation because we always write a gamma function x vlogs and it's got an easier expansion and then the idea is you calculate the coefficient x1 x4 well how do you do that well the very simple thing is you um i took the original i took the original point here to be at the top for the origin there's no a priori reason why I should pick that point if i pick this point here as the origin i do the conformal chance to make these two lines disappear and this collapses the other way but if you look at that the combination there is going to be different for the behavior of the compensation point it's going to be delta primed and delta pi likewise if i pick this point here the collapse will be here and here and so the discrepancy function will just depend on delta pi so we've got three different ways of evaluating this diagram but they've all got to be equivalent they'll all have a similar compensation function like this but a different expansion parameter because the argument would be different b f b which would be delta primed by delta tilde as I said and f c would be delta tilde and so three different ways plus all the fighters I neglected to write down in the slide brought together they should all be discrepancy and that's sufficient to be able to solve this to two orders that allows that allows people to write down sufficient terms in the expanse of leaving on a diagram to be able to evaluate for age of three and this is just an example I haven't written down all the terms but you see that you get the corrections involving obviously the problems themselves written out properly but you get these combinations of psi functions which is where they've actually come from in the results have shown so far the higher order diagrams come in in the corrections to scaling these are the two formulas to find the the first one here I represent to find all the diagrams have to sum up to one it's a very simple self-consistency equation and that fixes one of the internal parameters the second equation is in effect equivalent to the two-point function calculated as we did earlier but summed on the vertex functions and it's easier to calculate this the higher order diagrams have been evaluated years ago it's just a question of decorating with group theory but the leading order diagram is a bit harder to calculate which is what I've developed on it so in all of his formulation he sends off a point to an infinity for a three-point function and that in some sense is similar to what I do with sending the conformal transformation in I just make one point irrelevant and you left for two-point functions it's similar it's probably not connected and it's just I want to highlight that this morning and in anticipation where all of us going to talk and sorry all of I've put pressure on you but so what do you get well you get a result which is not the similar to lambda 2 in the sense structurally it's got Casimir structures obviously this now tags for the light-by-light diagrams come in in the calculation which is kind of nice because you sort of look at the structure of these diagrams through perturbation theory if you just focus on the expanse of these objects and powers of epsilon the other feature that comes in is that there's another new function which I call psi on this is a known function from early times when facilities work on this and it appears a non-carry symmetric theories and large animal of rain cube I've never quite understand the carol 70 aspect this but it's come up in two calculations one of which has been written up separately one of which I've known from the early 90s mid 90s from the rest of the model in supersymmetry but it's it's hardly complicated function to define if you take this two loop integral with decorations of well obviously epsilon depends but the d dimension v minus two and minus one and regularize the central line it's the derivative of this integral it's the derivative of the logarithm this integral with delta equals not the defines psi now we know if we've got a two loop function the two-loop two-point function if we have epsilon ordered from one on the lines adjoined at external points we can expand that in zetas if there's any decoration of order epsilon on the center line that's a problem and that that brings in non it brings in multiple sizes basically which is what derek and david have been we're looking at in the the mid 90s and this is the connection with where david and derek and I got working together in these problems in our paper david showed that this integral can be unlifted expressed as a hyper geometric function in d dimensions but the complication is it is because the difference is respect to the delta parameter here you have to differentiate the arguments the parametric arguments of the hyper geometric function to really define the function and that makes it harder to write it down as a closed form function of ones we know however what we did extract in that paper was the epsilon expansion of psi near two and four dimensions and it has zeta five three or zeta three five it has u six two and an old music but we now know it's all orders uh in epsilon and uh that's sufficient all right david and derek wrote down a an integral in uh in five four theory in their early work with um these multiple zeta structures but it was never shown whether it was realized in the field theory all of us and others have now found those in the field theory in five four this was another example an early example of for the zeta five three can appear in the field theory because we expand out in the parts of epsilon to find it in a certain order it's permanently early sort of what it comes in it is possible to value the integral exactly in three dimensions involves log two since eight of three of a pi squared and that's purely because of three dimensions you can exploit three dimensions and uniqueness of the diagram exactly all right time uh facility of detail no i haven't talked about checks on the results but those of us who look at not a billion gates if we've got a group uh generator inside interaction for say yang bills we immediately get qed from those calculates but just notionally taking t to be one uh just taking uh probably really mathematically on the signed limit of setting the generator to one uh well what that really means is you replace the casters by the corresponding object uh in the similar limit so the billion limit would be taken c f to be one t f to be one t f's the trace of t a t b uh obviously the the cast group cast me over to one but the billion limit has to have f a b c to be not such c a goes to not so i reproduce everything that was known before in the gross novel model by taking that limit from the results that presented today the other results that have been known as the s u 2 mott cons uh incident phase structure which is slightly different as s u 2 so it's still not a billion so it's three quarters and a half c a goes to two but the group cast me goes to five with 64 that reproduces what i had before and the more recent one which we've done is s o three uh which is uh got slightly different structure if you go and look it up on some table uh the group cast was 20 over three uh taking those elements reproduced what i did with my colleagues but also all these results agree with all that's known in perturbation theory and for most of these models it's either two-dimensional or four-dimensional but in this case the baxillian one we've only got four-dimensional results and we've calculated out of three looks in that and got agreement with the large n so they're both tally off which is reassuring okay now once you've got a generalized result you can just look at uh different group structures and so i've written down the three-dimensional results here just for the adjoint quotes for a bit of fun uh so uh put the firm into the firm fundamental uh the group cast me is obviously chains but the reason it's written this down is it shows you the structure uh more succinctly in three dimensions where the light by light diagrams have an effect and it identifies this term as a light by light contribution uh over involve this contribution well for either uh for either you go slightly different i kept this combination here because this tends to pop up a lot uh for the adjoint case it doesn't quite do a couple here but um if you're interested you put the numerical values in and see what it looks like but that's the kind of more general aspects that one can do with this formalism since doing this i've been looking at other universality classes with a different um underlying interaction i'm playing around with that and applying the same tools to sort of look at all the problems that were interactable it seems to be a really robust way of doing things taking a general lead group approach so just let me conclude um i've looked at a more general type of criticality class universality class for this fundamentalizing growth type interaction which opens up a whole idea now of looking at more general structures i can't emphasize enough on what's the color part is it for in form it's been makes life easy for doing the group theory uh you don't have to sit there with a pen and piece of paper and group generators around the diagram and it's easier to identify structures inside results and in principle all orders are basically by integrating through the epsilon expanse so i'll finish there um thanks everybody thank you john thanks for your nice talk if you're free to unmute yourself um and raise your hand or speak just up if you if you can um i have an immediate question to to go just one slide back um the three-dimensional thing you showed so so since you have this result for an arbitrary Lie algebra is there any special cancellations that you see for i mean for example here right i could see this bracket that you mentioned the third line from the bottom right so is you can imagine that there's a special situation with the spanishes and you drop the zeta three term or things like that i i did look at that enough to talk my head i think it doesn't happen okay this is actually a quartic in uh it's actually uh it's a quartic uh polynomial and uh and and the number of colors uh has to depend on if it's going to happen it's going to happen for low values of colors and i feel it doesn't i i should say i haven't looked at the exception groups yeah so i was just wondering since you have all this data i'm just curious if there is some something funny happening since you've raised that i i think of what i'll do is i'll look at it after it's ever come across and i'll put in the i'll put in the right up i'll make some i'll try to remember to make some come in the right up just yeah yeah thank you we have a question from david yes you've reminded me that in 1996 we did epsilon expansions about two and four dimensions to get towards three and there we encountered multiple zeta values but just out of mathematical curiosity i wanted to know what the epsilon expansion was about three dimensions and if you remember tolya kotikov and i actually did that because i knew that you would get alternating euler sums the same things you get in the um magnetic moment of the electron uh is there any use of those epsilon expansions that we did about three dimensions there could be david because um i i didn't put the slide up i had it in an earlier version of the talk i think you may have remembered this may have seen the slide before what you can do now is you can plot the behavior of the exponents in d dimensions from two to four and uh this is important because not only can you do that in large n now we're using paddy approximants across the dimension you also do in protobase two you know in a protobase two in two dimensions up to say four loops and the four dimensions down from four dimensions to four loops you can do what's known as a matched paddy approximate where you use the paddy information near two and near four to write it down and interpolate the behavior of the exponent across the dimensions and you can plot that you also in the function renormalization group they can plot the behavior of the exponents across d dimensions and they all have a similar sort of it's like a hump like a bell but soft set sometimes about three dimensions so what we could do with the epsilon span around three is expand in the neighborhood of three to see how well the um the maximum matches the shape of the other ones now it's not something it's until you mention it never occurred to me um and I can't remember what order in epsilon you went so it's obviously going to be um breakdown but what it could do it could influence the the perturbative expansion that people do from two to four with the epsilon expansion from a high order perturbation to try and sort of do some sort of global fit based on the data we know limited data we know from perturbation in large I mean it's it's not it's an exercise it could be worth doing but it's not something um I might have the tools do you immediately yeah thank you um I see some questions at the moment so let's thank John again here