 Thanks, so I hope you can hear me. I'm trying to look at you on a different device as well, but Let me look at that camera. Yeah, so it's great to well not be there, but be at least on this meeting So I want to give a little overview of a longer story. It started a few decades ago And then maybe in the end the second half talk about something that there doesn't know about so Let's start at the beginning. So 98 There was this this paper coming out where it is actually the abstract indicates already much of the contents, namely, They're then claimed to show that the process of realisation and capsules are up for structure in a natural manner. That was actually spot on a very nice very nice structure that that's appeared and this I want to talk about a little bit and about the So what happened quickly after that. And so with an icon and they showed that actually there was a relation with the big of decomposition. Actually, you could capture renormalization in terms of such a big of decomposition. And also from the start, it was actually clear that that there was a rich structure that should appear for for gaze theories. And this was already indicated at that time by papers were brought to St. Kramer and Kramer del Bugo where Electrodynamics was was used as an illustration. But of course, precisely for the point that what they are physically relevant and actually quite rich. And so with the paper called Anatomy of a Gaze Theory, which was written somewhat later. This gained a lot of momentum where there was this expression that actually you secretly saw in this In this on the whiteboard that was just shared with you. So apparently there was already discussion about this even today About the expression for the for the co-products on greens functions where you use this crafting operator where you insert graphs in one another. So I will explain it in a bit more detail later on. But there was this expression where you kind of say that the co-products sort of commutes in this sense with this crafting operator. And I'll come back to that later on. But that was claimed in a so-called Gaze Theory theorem to actually hold and also related and actually based on the identities that implement gaze symmetries at the level of the Feynman diagrams. And we did sloughnoff Taylor identities and there is a kind of back and forth between this expression and the fact that the sloughnoff Taylor identities are compatible with the co-products. And this was can be found in in many words at the time and also later so care needs, but also myself and with Derek and many other people that worked in this line. And I'm not even mentioning what happened if you if you look at the trees were also such expressions were were present. So that's what it does. For instance, in the case Now I was at the time of postdoc and that's where I met also Kourouche you just heard today. And so we had some nice meetings, I think so combinatorics and physics were happening. Two meetings and we gathered them in this booklet. And I started to work on the operative of Feynman graphs for for QED so quantum electrodynamics. And that's also when I made first contact with Dirk. And actually a bit later a few years later I managed to prove that these sloughnoff Taylor identities, they generate hope for deals, which of course is reflecting what Derek had in mind in this anatomy paper, expressing the compatibility of realisation with gauge symmetries. All right, so this was for me the start of a fruitful period of interaction and collaboration with Derek. So let me actually, since so to lighten up the the afternoon, start with some some pictures of the place we were supposed to be. So that's when I first or at least when I went there to visit Dirk in IGS snow. What's happening quite frequently in the year after again, this we don't see anymore. In fact, I think the climate changed so much that we don't see this often anymore. In any case, it's a pity we're not we're not there at this moment but maybe this helps remembering and actually this one is of course the nicer picture. So then a year later, they actually moved to Berlin, and which was of course a great opportunity, a lot of energy and momentum in this direction in the home world university with the form of the person ship off of Derek. And it was a great occasion and we spent many, many great moments also, on behalf of Matilda, we'd like to bring up this particular evening where we had a great dinner and of course we had great wine and a lot also and but we still appreciate this this moment so there was a great moment. Then, another thing I would like to mention and then, so don't worry I will come into details and so on in a moment but just a good location to kind of look back on this other nice place that Derek found or actually made put into shape which is the schools that that were done taking place at Le Rouge. Well, this is one of these kind of impressive things you can see out there. And, and I think also that for the group, I was having this was a great opportunity. So, and these are young people that really took advantage of the situation that we have these meetings on on particular the structure of quantum field theory and. So you see Matias are school fun and doing and you're doing is the young and they're all kind of, they really got much inspiration from these meetings. All right, so that's these were that was some time ago. Now, let's look at the, the stuff that that we're talking about. I want to talk about today is climbing graphs. I will start very generally where I say that that I have a set of vertices. This v1 to vk, and it could actually be infinite that is the core hop culture brought that was discussed by my blog and primer. And, and, and then there's a set of types of edges so it's like colorings on the vertices and edges and they have to be put together compatibility. I mean the pictures really tell you what you should be doing. So scalar five to the third. That's this first example, where you just have one, three of them to vertex electrodynamics that's where you have this one fault online appearing next to the edges, which you could by the way take with an orientation if you wish but I want to insist. There's this, this, this kind of rule where you put these things together in this way as you all will know so it's good to kind of get this structure together. Then there's your meals theory where there are essentially several types of vertices, even more like with ghosts and so on and there is also several types of edges. And they can also be put together according to these rules. So the types of vertices that we have. And then there's gravity. No, no, sorry, a particular gravity where you have like what is and the series of vertices and edges that appear and then we can construct graphs, such as the one written there. So, let's, let's look at what we're going to build with these. The, the, the, well, that's the start of the story the hopfaltz about Simon grass. It's the con primer hopfaltz about so what we look at this one particle irreducible graphs. And then I would like to talk about residues of graphs, which is kind of the remainder when you shrink all the interior to a point or so. Just giving me the type of graph so it's a vertex graph or an edge graph and that's what what's indicated with this. And what we look at is the free commutative algebra, which is generated by all five and graphs, given this set are where I also include trees. And then in there I could restrict to the one PI graphs with residue in this set are that I was given. Here's an example where you have actually one particle reducible graphs that is written here in this, in this picture. And that can be an M but it's not in age. Okay. And that's actually quite useful because so if we restrict. So let me let me first give you this map Lamla so actually the map is called Lamla and row in the same line so sorry for that. So consider a map from M to H tensor M, which I call row. And it's defined like this where you take this sum of her subgraphs. And that's, that's of course the subgraphs that are responsible for possible divergencies in eventual amplitudes. And the quotient by this subgraph and actually it could be several so it's addition union. That's just by shrinking that subgraph to a point to a vertex or an edge, depending on the type of graph. And so this is like the, the co products, except that now I define it on M. But if you restrict it to this age. So if you restrict row to age, then it's actually the co products. So for the hopf algebra age and concrime or hopf algebra where this co units is just given by something not really only on an empty graph. In fact, this whole package of M and age comes together by saying that M is a left age co module algebra. So actually this, this hopf algebra acts on M in this way or it co acts rather by by the same formula row. So it's a very natural thing to consider because they are defining graphs, and we want to renormalize them. So, some examples that you've also are also very familiar with this formula, where I shrink these, these boxes on the left hand side and I expand them on the right there's this primitive part and then there's these, these red subgraphs for instance in that line you find that there's like these two pieces, and they're disjoint union is the one that I'm taking care of on this on the third term on that line. And then there is also the one in which you take a larger graph in the blue box, etc. In the row, if you consider that that map on this kind of glasses. Then you find that there is not so much a primitive part, but it's actually, you only take into account the one PI graphs on the first leg of the tense price so that's why it's a little bit different in that case but otherwise it's the same principle. And of course if you try to shrink a graph which is not in your original sets of vertices or edges then you better not do it. So, that's what, what's written in this last line. Right. So let's refresh your minds on how to get renormalization as this kind of beak of decomposition in G. I will not go too much into detail on this beak of property but it's, it's what I would like to say is that if you take a character use that which is for instance given by something which is like a regularized find my name for Jude. Then you could define a character on his whole project CZ, depending also in this parameter Z is regularizing parameter, which is defined in this way where this T is somehow kind of projecting really onto the part which is responsive responsible for the for the urgency. So it's like the projection on a pole part for instance and then what you find is that if you kind of systematically subtract this part, like I said in the counter term you counter this divergence, and you end up with something which is the renormalized value, or Z, especially finite when you put that equals to zero. So that's the good old story of this, this procedure of renormalization and that is just the choice for convenience, the dimensional regularization but many other possibilities exist. All right so let's look at this. One remark might be said that what is CZ these counter terms are defined on H. But RZ is actually defined as a map from M to C so it's actually for all fine and graphs that this procedure works but counter terms of course you should only take into account the one PI graphs. So that's the usual story of renormalization. So this is quantum field theories and is quite general now for any type of quantum field theory, any type of roadie season edges. But here is the interesting story of gaze theories which actually that's from the basis of the work I did and also the work I did with Derrick is that if you look at at first of all if you look at the greens functions. The one PI greens function so they are actually given like this. So there's some kind of form factor, which is the thing that you would typically find in a Lagrangian. And then there is a part with which it's multiplied in order to get the full kind of the quantized contribution to the renormalized value for coupling constants or for what for the usual greens function in quantum field theory. So there is GRs that I have in there and their elements in the, in the Hopf algebra. So they're formal series where I take into account all those graphs, just sum them all one PI graphs with the given residue, namely the I or EGA. And then there's a plus and minus of course and then there's the symmetry group that you take into account. And if you think about what happens for gaze theories it's quite important that there are certain identities between these form factors because otherwise you will never have such a gaze symmetry because that's precisely where the notion of, I mean, where gauge symmetry groups are acting in making sure that these relations should hold. So my question is, actually, it's a desire or demand almost to say that renormalize ability for gaze theories really requests that they're that this type of relation kind of persists also at higher orders so for any greens function they should hold. So if you look at this expression over I say on the left hand side here so this is this four point function can be expressed in terms. And the sloughness Taylor identity should hold where you say that the G the for loop times the and the edge G is actually the square of the trivalent vertex and and these type of identities. They're supposed to hold in order to guarantee renormalizability for these theories. And then it's important that you know that they are actually they hold at the level of the whole project. And that's where we kind of were first motivated by looking for an expression for the co-products on these GRs whether or not they're actually compatible with the co-products. And, of course, you would like to impose these could these these identities at the opfalture graph but it should be in such a way that what you end up with is still a opfalture graph so you're going to just impose them that should be kind of compatible with the co-products. This is what we'll, we'll uncover and recall. So, first of all, a little bit of structure of of age. So there are several gradings there's a grading by loop number is the Betty number decomposing age into these pieces HL and QL is the projection. And there's also a multi grading by number of vertices where I kind of count the number of vertices of a different type. So vi. And I correct for some some possibility that the residue is equal to that vi, then it actually becomes a multi grading which is kind of makes sense at the level of the opfalture graph so it's a grading on the opfalture graph. Composites like this and there is a nice relation between them, as usual, we kind of can express the loop number in terms of these other DIS is not a grading in this way. And also note that these are examples of connected opfalture rise where the zero of order is is actually just the scalars. So this is what we, what we have this kind of package of ratings which is useful later on. If we look at the next sample of say scalar fight to the third, there's only one type of vertex and one type of edge let's see what happens if you compute this corporate on these greens functions. So then what you can do is you can, you can look at the elements x, which is a combination of GV and GE in this way. And then together with the GE is degenerate a hopf sub algebra in H. And that's precisely because you can show that the corporate act on X is actually just expressible in terms of acts, restricting to each loop order. And the corporate on his GE can be expressed as well in terms of G is an X. So it means that the GE and the GV to these two greens functions they generate a hopf sub algebra in H. But what is the structure of that hopf algebra, if you dualize this. Remember, they're just commutative hopf algebra so there's a group which is dual to this. Then you find that these are that these are just formal powers here which are invertible in some lambda. And there is a cross product with the diffeomorphisms and in fact what happens is this is, if you take a character on this hopf sub algebra which is supposed to be the element in the group. You get an invertible invertible formal power series by by this expression. And then there is a formal diffeomorphism which is defined like this. So it's, it's completely kind of the way to understand this is by looking at these are diffeomorphisms and this is very formal power series. So what happens to be the structure that we will find in all cases, if we would have such something like sloughn of Taylor identities. So let's look at how that works. So the structure of of age in general is like this so it's. And then there are these vertices as before v1 until vk. And then there are these edges e1 until e n. And for each vertex, we define such a like a charge so x v is defined like this where you just take the greens function corresponding to that vertex. And you divide formally of course, using the loop order for instance, to make sense of this expression. And then you divide up by the, the, the edges or the greens function for the edges which are actually connected to that vertex, and then take a square roots as before, before we had three over two, because it was prevalent at vertex. And that's what we ended up with also the one over three minus two is actually just one. So that's, that's the kind of the charge of the expression that you would get for for five to the third theory but this is now in full generality. You can ask the same question so what is the corporate on greens functions. And you find that actually this. There's an expression which you can show that's an exercise in combinatorics it's rather lengthy and what one can do it. The corporate on this gr is actually given like this is gr. And then there's these powers of x, the x v's appearing. And then there is the part of gr on the second leg of the transfer product which is of multi degree and one to nk at that part. But it's actually, if you will take into account the multi grading, then it's actually fine you see that, well, this already generates for me. Something like, like a sub algebra. And you can even say that the delta of x v can also be expressed like this we have all these terms x v one appearing again as before. So nothing really changes here. So that's the structuring of the previous formula. But what is what it says is that you have the multi degree is necessary to get these hope sub algebras. But now suppose and that's the point which will come in the next slide is that when, when would I get something like a sub algebra at each loop level because that's kind of the physically relevant structure and that you restrict yourself to, to x v, or whatever in gr at a loop order L and not at this multi degree because it's not really physical to count number of vertices. So that could happen, precisely when all these x v's that you see here on the on this on the screen, when the when the actually eyes are all identical. So you have this formula relating the, the multi degree is the eyes that I had before. It's related to the two L, and then you actually find that the, the exponents of these x v eyes, they combine to give you something like two L and then there's this other x v so it's two L plus one just as with five to the third period. So you find that that's actually, well, quite important to have, if you would want to have something like. So that's the expression that indicates. So I don't know where I, where you started missing me but just start talking at some point in it. You try to catch up. Yeah, so all these x v eyes when they're equal, then actually you find that these exponents can be combined to be related to the to the loop number. And then you would actually find that that if you project on the second leg then you're projecting on to. So that's what's happening. And this would indicate that you have this compatibility with with these, you have this closed expression whenever you have these relations x vi equals x vj so let's do that so. For example, you get this up so about your age prime, that's in all these different multi degree so you have to take into account different couplings for all the vertices. But then, if you would impose that the x vi and the x vj are equal, then you actually are reducing it. And the fact that the reason I just explained is that all these exponents combine and conspired if you twice the loop order tells you that this is a whole five deals so it's actually closed under that expression. And, and you can find it that what you end up with is a whole. The quotient is a whole photograph, and then you can conclude that the dual group is just the one that we had for five to the third. Just formal power series in one variable acting in different, maybe for the different fields. So that's way function realization for the end different fields, and then it's semi direct product with the different morphisms. Okay, so that's, that's kind of something that that that that we showed some time ago. That's, well, yeah, that's just a name attached to it. So, they're the sloth no terror entities but I would like to get to the gaze theory theorem. So, let's look at the, the whole shield homology for hopf algebras, not to be confused with the whole shield homology I will be talking about a bit later for algebras this is for hopf algebras and then you use the co products. To kind of to find a structure of a complex first of all but but let's go slowly so let's look at the bi algebra age. This is an H by co module so it's some some left co action and a right co action which are co community, just co everything. What you know from from group actions and then you have role role are these are the left and the right co action. So what we will be looking at is to like code chains in this case. There will be linear maps fine from M into the n thoughts tensor product of H. So, on that, we can define a map that's the whole shield co boundary map B, which gives you from because from CN to CN plus one. So what you do is you take the co products in between the different components that you could that you have in H to the end and on M you use the role role and the role are. So, this is what you end up with. So, after five, and so after evaluating with respect to fire what you end up with is something which is in H to the end, on which you can apply in the ice term you can apply the co product and then it gets to age tensor and plus one. So this co product in the ice factor and then as I said the role role are used as the first and the last term, and the co associativity of delta of the co product actually implies that B squared is equal to zero system, conservation of the minus science that you find over there. And that means that I can you find a co homology groups, which are the whole shield co homology of the bi algebra or hope for java H with values in M. And so these are precisely the co monology of this complex. So, let's see how this is relevant for what we've been doing well I actually saw already an expression that is very similar to to what I'm about to tell you now, in fact it's the same. It said, if you consider M. As H a co module over itself, and you take as a left co action just the co products, but on the right you take the co products composed with the identity dumps co units. We had a whole shield co monology group, which is denoted by H H epsilon, which was already present in the early papers of quantum primer. It was actually relevant for these. When you look at at root of trees that that's actually the case that's the structure underlying this. And then you find that that to be, for instance, a one co cycle or shield one co cycle, it means that delta of Phi, that's representative Phi is equal to identity times Phi, after Delta, plus five times one, what this means just plugging in a one. The, the picture you saw on the whiteboard in the little break was precisely telling you that this be plus gamma for primitive graph so the grafting operator is a whole shield co cycle of degree one. So that's precisely this formula over here. It tells you that so the grafting operator just takes a primitive graph gamma, and it plugs in graphs. And actually that's the, the argument is X K R, which is the greens function with respect to arm for the vertex or the edge R, and then to K powers of X V. And the point is that this actually holds. Whenever you impose these X V one and X V to all these, these churches to be equal so you have to impose the sloth no teller identities in order to show this. So the content of the gaze theory theorem is actually precisely that relation between these two, two aspects so this hope for deals on the one side, the outbreak structure and this this this whole shield which is at place at this in this theorem. So, now of course this extends when you when you sum over all primitive graphs of a certain loop order and a certain residue, and you still have that property because it was just a linear property. So you can find that that you can actually show the gaze theory theorem that they're opposed back in 2005 in the preprint and pretty here later. It was that in the quotient hopper algebra. First of all, you find that you can generate all the greens functions by just doing this grafting all the time in a kind of in the in the right way. So this property that I just showed you for primitive graphs gamma that extends to be plus KR so we're certain vertex type and certain loop order can actually do that. And you find that you actually, if you look at one and two together, you said it's well then I can just plug in gr at that place at loop order K. And the co product on that is just expressible in terms of the thing I started with, and the polynomials in G on the first leg but times the greens function for the same type are same residue, but that the lower loop order. And that's precisely the, the essence of this, the stories that if you impose the sloth no teller entities, then you have that property of these of this whole shield co cycle. And then you actually find that the hope for to generate it by this GR is actually the hops up a device is actually makes sense on this GRK. All right, so that was the story as we, as we had it. So this is this can be written or read in the in a paper I wrote a little bit later where I just gathered and you can also get all the references to all the papers. But I would like to maybe tell something as I said that that there may not know, because this he all knows for very well for a very long time. So, let's talk about the unexpected influence on non cumulative geometry that that Derek had even though it's not really unexpected in the sense that he was in beer so it was kind of hard to not have that. But what I want to say with it what I want to tell this is that that that as far as I remember I have never seen Derek working on on non cumulative geometry in the sense of spectral triples or so. And with a strong interest, of course, in these developments, and for me it was an inspiration and motivation to actually uncover the field theories in in our community of geometry as they appear, especially what is the real normalizability properties, how are they defined and are they actually physically viable in the sense. So let me tell a little bit about this. So that's a story that one probably should not be telling in beer. But but let me still do it since we're now anyways, online so. So this is this notion of no community of geometry is based on an on an algebra of coordinates to replace kind of spaces by algebras coordinates could be no commutative. And we take a generalization of the rock operator and D where you actually just forget about all the kind of geometric structure that was used to define the director. But only remember the analytical properties like self adjoining some some compactness of the resolve and so all kinds of things that you can derive. So you can actually just isolate and abstract and work with that by itself and then allow a lot more and also some more in the direction of non commutative geometry, no commutative spaces. So these algebra and operator they come together by in a helper space where they act as operators. So the algebra is acting as bounded operators and the operator is an unbounded operator with suitable properties. So let's follow on the details on this because that's a little bit and outside of also the theme of this meeting but what I would like to get to is how these physical applications come about. So, well, this is good to keep in mind. So that's just an example where you take like functions on our for you take spinors on our for on which you act. So this is a smooth multiplication, and then the rock operator acts on these, these spinors as well as it was defined like that. These smooth spinors and this extensive and unbounded operator on this helper space. But the kind of the origin of field series or gaze series is the following. And what you can do is you can take your, your D is the rock operator and you can fluctuate it. And these fluctuations can be driven by the algebra. And this is the only thing kind of the ingredients I want to take is that if you, if you look at this. Like V, it's written like the series, AJ, and then the commutator BJ, then what you find is that. So the AJ and the BJ their elements in the algebra. And what I do is I construct something like a one form with them by taking this differential with respect to D. And what I do is I just add them to D is such a way that the result is still self a joint. And that's kind of the sort of the V is supposed to be thought of as a gaze field or potential that you add to D. And I'm sorry. She also tried to maybe sort of that. Yeah. So, here, the, you get this, this fluctuation of these are the geometry, the rock operator fluctuates like that but now I want to look at action functionals. And what I will be looking at is the following the spectral action was introduced also about 98 so it's jumps in and con that it is 97 98. So it's almost in parallel where you look at the physics that you could try to that you could get from this functional by looking at it as a functional on V. So this is what I, what I took serious and I looked at it for several years also several decades even to look at the structure of that, that action functional. And what what I in particular try to understand is the structure it has, what kind of renormalizability properties it has. So, first thing that that I actually developed while visiting blue is is the following formula. So you can look at the, the expansion is like a Taylor expansion of this, this action functional as a functional V, where you where you end up with the formula I wrote here we take contour integration. Also the details are not so important it's. It's mainly the structure that you get when you look at these these these resolves and set minus D to the minus one. Then it's kind of multiplied by V and that's taken to the nth power. That's actually my Taylor expansion of the spectral action functional. And the dependence on F is this function F, which is like a cutoff function is precisely in this in this contour integration. What I'm doing, I'm currently doing with a PhD student turned from new land is the following is that we try to understand a little bit better, the structure of these terms. And the first thing we're doing is for well let's let's write this in terms of so called involved in column just brackets so V1 V2 until V and they can be put in this is contour integration under the trace. Then you can look at, say, you try to define a whole shield code chains which right now are for the algebra structure. So what we do is we take. It's like n plus one linear functionals, so it's multi linear functionals in n plus one variables to see so that's my fee my fine and and so I put in these elements in the algebra. And what you find is so what I used to define them was exactly this committed with D in there. What you can show precisely because of this formula and the fact that I have these resolvance of D appearing there that you find that if you take now the whole shields differential but now for the algebra structure so it's not using the co product but the co product in the algebra, then you find that first of all that bio be of fire and is equal to fire as one for odd and, and for the even ends, we find that fire and are actually a whole shield co cycles. So that's just also if you know a little bit about this is not so so much of a surprise but it still isn't very nice structure. So what we can try to do is understand a little bit how this, how these terms can be written in terms of these finance. So let's do that. One of the first things that you find is that the first few terms in the expansion. We find that that the adb, which is just phi one evaluated on a comma B, which is written like this in terms of an integral over the universal differential one form a so it's adb. It's universal in the sense that that any other differential can be derived from it or as a quotient it appears in particular the one we take commentators with D. So you have this adb universal differential one form. That's a very much algebraic object. It's integrated against these five one, and it's integrated when you square it against fight two. And these are the terms that appear in the expansion. So we take where you just look at the terms that that pop up the left hand side is very simple, but they are precisely the ones that kind of give me the Taylor expansion of this action functional. And what you find is that that here is the recollection of these terms. I would say the first few is that if you take, say, look at the ones which are with respect to fight two, we can rewrite them kind of taking into account the fight three or five one as we said before is that little be a fire one is actually fight two so you can actually manipulate these these ingredients and this is what we did. And you find that that you can combine these terms in such a way that that you can recognize something like a curvature. So it's like da plus a squared, which is the curvature of a. And then there's da plus a squared squared so the curvature squared against five four. And this actually continues so what you find is that, first of all, you get these integrals of the curvature and the curvature squared to the third term to the third power is integrated against these even field co cycles. But then you find that the other ones. So the a, then there's the other term appearing which is a da plus 238 to the third, which is also ring a bell what kind of expression that is. And then there's another one integrated against psi five, which is a da squared plus three thirds, three house eights to the third day or doesn't really matter what it is unless you know what I'm talking about is that these terms is these forms that appear here which are universal forms are actually well known. So the first one are just the young mills terms in the sense that their powers of the of the curvature, they're integrated integrated against even whole shield co cycles, five to five four and five six, but the other terms are the structure and Simon's forms of order one, three and five, and they're integrated against all cyclic co cycles. So in fact that's the structure that is a bit more sophisticated than just off shields. It's actually off shield would correspond to something like taking integrals and cyclic co cycles is something more like drama homology. So you use something like drama homology if you wish to, to integrate against. Now, the, if you look at the scaling there in part of the spectral action this, this this was already so that's a part of this was uncovered by Shamsen and con. But what we found now is that this actually persists not just at all levels of the spectral action but also at all orders in the perturbation. What we find that's the structure we have is that, first of all, we define this turn Simon's form of degree two and minus one, which is given like this. And the final results is that if you take this F is da plus a squared a take this turn Simon's forms of this degree. So, then you conclude that the, the, the close expression that you find for the Taylor expansion of the spectral action is just in terms of powers of the curvature F to the K plus plus one. That's integrated against this, even cross cycle and then there's this turn Simon's form which is integrated against this cyclic cycle. It's a simple and basic form, but that's it so there's no other term so all the examples that you could generate you could think of. They should all be of that form, what do you kind of the freedom is in the form of these fine side which depend of course on the I started with. So that's actually what I want to end with is that so this is the same expression. What it tells me is that that is very simple structure of the spectral action way just encounter these two forms. This, this is the moment to to look at to a serious study of the gauge structure of this field theory, because now you may wonder well this is kind of restrictive, but you also know that counter terms are restrictive so when you look at BRST symmetry, it's possible to kind of tell me what should be the kind of the BRST invariance of the counter term or the gauge invariance of these terms, can I actually incorporate them in terms of the structure that I started with kind of ever running or any variation going on in this whole shield in the sickly co cycles. So that's what's what we're doing right now. But for, for this moment today. Let me just think there for this continuing inspiration also in this, this field of now committed to this. Thank you very much. Thanks, Walter. Thank you. No surprise indeed. I have not expected to to venture so so deeply into non commutative geometry. I honestly speaking. Finding something you could do with it. So do you do you see a lot of co ideas or anything. And you just go at BST do you see a similar structure coming up. And I think the connection but so for instance BRST at the level of Lagrangian. And the fact that you look at flows which respect that invariance that's of course, telling you indicating a subgroup. That's the kind of the understanding of the, the ideals, as you know. So that's maybe a link but for now I mean it's so there's always this this thing about BRST invariance taking place at the classical Lagrangian level. Whereas it's much more interesting actually to look at it. I mean, in fact, this is what I learned from you probably is that that you should look at this more from the, say, at the level of on the open space level. So that's why you should be looking at this. So, but that's all to be uncovered. So for instance if you look at so not committed to geometry and it's. And it's kind of, it's attempts to move this into into into a quantum theory quantum field theory. Then you could say something maybe about the fermionic structure, we tried to do some kind of Hilbert space quantization of that. But then the bosonic part is always the issue so you end up with this Lagrangian which is very difficult for the bosonic components. And then you have to deal with that so that's still open to. Okay. Thanks. I found the question since you mentioned BRST. Recently I've seen a lot of work on on formulating BRST and kind of L infinity algebraic language. And I was wondering from this whole shield and co cycle part of you is there any connection between these two this does one mean something in terms of the other we have a feeling. Yes, so there's actually they're closely related so but but that's even before infinity because that would be more on the kind of bv side I would say. So, but I will call it and then you have that connection. But just if you look at BRST then. So what you usually do but this is a Weinberg style so it's it's it's a bit older is that you look at. So BRST invariant functional so you try to kind of compute the BRST homology. But then in certain cases you can actually read relate this to lead algebraic homology, which you can typically compute. And then you can quickly kind of conclude that the counter terms are of that type. But the algebraic homology is easily related to off shield common of the universal enveloping algebra. So then you're home. But it's not the same type as I'm discussing here so this is a different algebra is the, well, I mean, it could be a similar algebra but it's really the ghost sector that drives this part of the story. So it's the algebraic homology on the on that part. Thank you. Thank you. Thanks. I see no further questions at the moment so so that's thank water again.