From Complementations on Lattices to Locality

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From Complementations on Lattices to Locality
(Or from renormalisation to quantum logic)
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A complementation proves useful to separate divergent terms from convergent terms. Hence the relevance of complementation in the context of renormalisation. The very notion of separation is furthermore related to that of locality. We extend the correspondence between Euclidean structures on vector spaces and orthogonal complementation to a one to one correspondence between a class of locality structures and orthocomplementations on bounded lattices.
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so thank you very much uh karen and eric for the invitation it's a real honor for me to be here on that very special occasion and uh first of all let me say happy
we should have been in france so i should say by the universe and since it's early morning very early morning for some of you this is going to be a leisurely talk not
too serious but it's going to take us along winding roads from complementation which are defined shortly you can think of the orthogonal complement for the moment as a guiding concept to locality relate in relation with locality in quantum field theory and i discovered that in fact this takes us to quantum logic which is not my field of research at all but we have had to look into the literature there and discovered that things we had thought uh we discovered we'd proven were new were in fact known in some parts of quantum logic and um my um motivation comes from renormalization and there i got a lot of inspiration from the work of dirk and uh with alan cohen and i'll go back to this this is joint worth with pierre clavier liguo and binja so happy birthday and now let me go to the motivations so renormalization and locality i'm going to put that in a very concise description which of course is over simplifying a complex very complex picture so i'll have i'll tell you later what locality means uh we'll go back to that it's just a binary symmetric relation so we have an algebra this will be this binary symmetric relation i'll come back to later and a product you can think of an algebra feinman graphs with a concatenation of graphs trees also with a concatenation of trees well forests i should say building concatenating trees building forests and cones which uh so these are convex polyhedral cones which were in fact the starting point for our study and now i'm going to be vague at this stress at this stage just saying there will be another algebra of meromorphic germs at zero i'm not yet specifying whether one variable like in the uh algebraic because uh factorization of uh dirk and kun or as we will opt for very shortly the algebra of multivariable uh meromorphic germs at zero with linear poles so here we'll have several variables now this is where we land this m which we haven't yet completely specified but we know it's something like meromorphic germs at zero so it's it's where the singularities the divergences arise and we have a morphism phi from this algebra find my graphs pharisees cones into this algebra of meromorphic germs these can be given by five man integrals regularized of course here i'm sub-intending because i land in meroporfic germs that i'm using i could be using dimensional regularization or some other regularization i have functions so they have present poles but i can define this find my integrals or branched zeta function or conical data functions i won't say too much about this because this is not really what we're going to talk about today we're going to stay on the right hand side where these metamorphic germs are what we're interested in of course we don't want metamorphic germs to have to measure we want numbers and if we want numbers uh we'll have to replace this target algebra by c before that this is where this locality which i haven't told you very much about but you can now maybe guess you see it reflects locality in quantum field theory in as far as it says that if two iis are independent for this relation then their values the value of the product this concatenation the value of this concatenation will factorize and will be a product of the evaluated files over each argument so this will go back to it's a kind of separating device it tells you these events are far apart and hence this is why there will be no extra term coming from whatever could happen between them what we want to do is as i said we want to go from m to c and that's what everybody wants to do to have numbers and so here it's a black box all i know is that when the ais these feinman graphs trees cones are just concatenated without further interaction then i would like this value here to factorize so i would like to keep this when somehow we extract from these meromorphic germs some number so this is a black box okay so how do we how have what have i learned from dirk and alan khan i've learned that if you have as a target space m one variable so this is typically a result of uh working with dimensional regularization then um you yes there's a question no okay so um you want to separate the poles you you have your function phi we had in the previous uh your map five phi of a you have an a in here phi of a is a metamorphic germ what you do you want to separate the holomorphic part which i'll write in blue from the polar part which i'll write in red and if you do that just naively it won't work namely this multiplicativity property we had here down below will be spoiled so you have to be a little more clever and very clever indeed was the idea of introducing co-products which sepa which kind of undo whatever mistakes you might have made having naively taken this uh holomorphic part and evaluated at zero to get a number in several variables um so here this uses a coproduct so it uses a lot of structure on the first algebra on a here we will focus on the target algebra m and we will want to separate it remember it's from our point of view we'll take several variables so we'll have lots of variables to deal with and this separation is not at all um straightforward it's in working out this separation that we're that's what we're going to focus on today how what are the mechanisms behind this separation in order to be able to build this renormalized map we want it so when you look at logo expansions in one variable you know what to do there's a holomorphic part i'm at a point zero zero will be zero and then you take uh the holomorphic part and take out all the poles polar parts okay and of course if you do that for this big phi and just take pi plus compose with phi as a five plus you won't have a multiplicative property
i mentioned but uh so what we want to do is generalize what looks like a simple artifact space splitting this and this is minimal subtraction scheme one could say but we're going to do it in multiple variables so for that we have to understand how to separate the polar part from the holomorphic part and this is what the complement mapping the complementation maps will help us to do so please don't hesitate to ask questions along the way so a good way of separating is considering orthogonal objects when you have some orthogonality around them around at your disposal so i'll take a vector space i could take it infinite dimensional but let's not worry about this let's take it finite dimensional and this this object here g of v is the set of all linear subspaces of v and let's take an inner product on v very simple facts what does it define it defines a an inner product well it's defined as a as a bilinear form and so it is a symmetric relation i'm assuming i'm in a real case here and so it defines for me a binary symmetric relation u is orthogonal to w if and only if q of u v u w is zero for any u w in this respective spaces a very tightly uh related object is this complement map which takes you as seen here as an element of what is later going to be a lattice the lattice of linear subspaces taking it to another one what does it do it takes it to the orthogonal and of course the true are very much related u is orthogonal to w if and only if w is in the orthogonal complement so this is the polar set which is here the polar set of u or the polar space is all consists of all the ones that are that all the w's that satisfy q u w equals zero so this looks like a very simple thing but for us you see this this u orthogonal is related to the s symmetric binary relation represented by q and so this is uh this is on this side and um now how do we get back psyche from the max from the uh polar set so the polar said with this one and this complement orthogonal complement will be the maximum for uh a post a partial ordering given by inclusion so this is how we can go back and forth between the two and why are we interested in orthogonal complements on it looks like you know very simple first year um material why do we why are we interested in this because we will use them to separate polar parts from holomorphic parts in our metamorphic germs and remember that was what we wanted to do relative complement maps are ubiquitous when you start looking at them uh closely you see that they arise in many uh ways and not surprisingly in co-products and not surprisingly in the one that was introduced by dirk and alla when looking at feinman grass or rooted trees because what does such a comp uh how is such a co-product built it's built from separating i don't know whether you see i've got things in them in front of what i'm trying to show you so you're separating x uh yes so i think there's a so y so you're separating y from the complement of y in a in a certain element x so remember we have a post-set structure on feinman diagrams to be a sub-graph or find my graphs to be a sub-graph off or to be a sub-rooted tree off and if you think of the co-products the way they're built they use a complementation they take a sub-graph a sub-tree and separate it from its complement which i won't describe now for each of these of these co-products you have to give a precise definition of what the complement is surprisingly you also see those complementations as an abstract um notion something which takes an object to a complement in a meaning that they build together you can rebuild this the space from those two parts this arises uh in non in equivalent geometry or toric geometry uh so that's where i first learned about it in some paper by promise harmon gahufalidis about the euler maclaurin formula on the convex polytopes because hidden behind there there is such a coproduct so the complement map is the transverse cone to a face of a cone or you look at cones in fact rather than polytopes and your complement map is taking a transverse face to a face of a cone which i don't want to dwell on here so all we want to remember from here we need some kind of rigid complementation to separate polar parts from homomorphic parts so because we want to be systematic and that's in order maybe to hopefully at the end get a better grasp on um a kind of uh over um renormalization group or maybe i should call it galwa group we want to be systematic and see which symmetric binary relations define a reasonable complement map and conversely how do you go back and forth so we want to generalize this this relation remember we had orthogonality which was seen as a binary symmetric relation and here a complementation which sent a subspace to its orthogonal complement so this is one way and the other way um so what we wanted to do here so we did it both ways and what we want to do here is forget about the orthogonality try to find conditions on these symmetric binary relations in order to relate them to some complementation we have to define what that means and the natural setup is surprisingly low uh lattices and because we're having this locality with we're um progressing with this locality a lot all along the way it's because we want to keep in mind this constraint of a factorization over independent uh factorization of measurements over independent events but then if you look at the literature you see that under completely different names this was studied for other reasons in relation to quantum logic
so so here what we expect is i can't see what i'm showing i hope you can i have things in front of it that you is um independent we call it of w you can think orthogonal if and only if u is in the orthogonal so in psi t of u sorry w is inside t of u so this is a generalization of what i wrote here we're just replacing orthogonal q by t by this t and this by u t u t meaning we take all the ones that are indeed independent of uh the elements in uh the the elements in u and here we do the same thing we have a psi t instead of a psi q and here the maximum the maximum will make sense because we have a postset structure and a bounded structure and we'll take here the a polar just the one i just described so this is what is written here that's what we expect and now we have to make sure that this makes sense in this very general locality lattice setup just so that i convince you that orthogonality is important in laurent expansions let me quickly tell you why it's everywhere because you see we're looking at this kind of function in several variables the allies here are linear forms in some z1zks and the big allies as well they're linear forms and that's why we're going to talk about germs at zero with linear forms h being a holomorphic at zero so that's what is written here example very simple example and so now we ask the question what what happens if i said said one equal z2 equals zero what do i do at this stage and we want to addre sorry we want to address this question directly and so for that we define the uh the dependence set of the uh linear forms entering in this function f this gerb to make sure we'll be able to decide whether the linear forms this f depends on are have nothing to do with linear forms another g let's say depends on so this is what we do here you see we say i'm using abusively the same notation as a as a orthogonality a usual orthogonality um writing f1 has nothing to do with f2 if the variables they depend on have nothing to do with each other in the sense that the spaces generated by these linear forms are orthogonal so q here is fixed so for example if i take the canonical inner product on r2 i would say that this z1 master set 2 is independent of z1 plus f2 you view them as carried by the vectors e1 minus e2 e1 plus e2 where e1 e2 would be an orthonormal basis for q and then we have to say which ones we want to keep which ones we don't want to keep now we have a general notion of germ and among those i want to uh throw some away and i'll throw those away which for which the top part has nothing to do with the bottom part so which are irreducible so these are just recalling what setup we're in and now that's the important condition the top part sorry the top part is also the space the dependent space generated by the top part is independent of that of the bottom part and then i'll decide this we throw away if this is the case we'll decide this is polar and they generate a subspace and to be very precise because we're going to have to use um we have to go to loha expansion we'll have to look a bit more closely at all these expressions and keep track of the kind of linear forms arising in the polar part for that we use the language of cones that's where we came from but in fact it enlighted us on how to deal with such meromorphic germs and we'll say we'll have families of cones because you see we have several of these s s js i'll call them soon several polar parts and we have for each of them a family a set of linear forms and these linear forms will organize in cones they will be the um the um sorry the the edges of the cone that these will form the edges of the cone and then we don't want overlapping and this is uh we don't want these codes to overlap otherwise we're doing over counting so this is this condition properly positioned which is a bit technical and then uh so that's what we require of the polar germs that will arise when we look at one of these general metamorphic terms at zero so uh this was um a result that we proved using inspired a lot by uh nicole berlin and michelle verne's work uh that using an inner product and the complementation we could separate m into m plus and then minus q those are the poles the ones for which the numerator is has linear forms orthogonal to these those of the denominator so here are the technical conditions uh just so that we have uniqueness holomorphic up there so this will be homomorphic so what are we doing we're separating this metamorphic germ into a holomorphic part at zero and a black box but we can look into this black box this contains it's like the dustbin it contains all the polar germs and to make sure we understand them we keep track of the cones the supporting cones and ask that they're simply short so linearly independent operators so be careful um the this decomposition is not unique and 1 over l1 l2 is can be written in two ways and this is why these supporting codes are very important the first one has supporting codes um with edges given by l1 l2 whereas the other one has different supporting cones l1 l1 plus l2 so this is very much related to uh double zeta functions in a sense you're splitting the upper quarter plane into eighth of planes uh using the uh the um line y equal x are there any questions okay so this was to show you that orthogonality is very important so use it where did it come up just so that we're uh on the same line of thought it was here so it looks kind of harmless but that's what decides for
the splitting and hence for what we're going to keep because the in the end we would like to say ah what we want to keep is h of zero we throw everything up away and so it's a it would be a multi variable minimal subtraction scheme okay and but knowing the lower expansion gives us much more insight and we will use that to define not today but we're using that and that's why we're very interested in a generalization of this theorem to describe a galwa group um of meromorphic germs um modulo of transformations of meromorphic germs that are the identity on the holomorphic part so you see this was proven for q and this special inner product which gave us a locality we would like now to prove that in a much more general setup namely replacing this by a locality and replacing this complement map by the if i call t the locality psi t the com the associated complementation so that's why i'm looking at orthogonality as a locality relation because i want then to go one step up from orthogonality to a general or relatively general locality relation so what is a locality um so g of v i remind you was the set of uh finite dimensional or closed linear subspace of v and there is a natural partial order uh to be a linear subspace of or closed if you need it so for example um yes so what is a locality relation a symmetric binary relation and you see orthogonality gives you one on g of e because it tells you if two two linear spaces are orthogonal or not and so this gives you this binary uh symmetric relation again uh on you see here we used this binary symmetric relation to this to declare or to decide whether two functions two metamorphic germs were independent or not we even abused somewhat of the notations using orthogonality so okay so the lattice g of v that's the one we're interested in what is it it's a poset what is a lattice it's a poset a partially ordered set with a joy and a meat so for our lattice this will be sum of two vector spaces this will be the intersection of two spaces um usually i mean those operations are usually associative and and they should be if we have a nice compatibility with the partial order they should be compatible so monotone one says isotone in that area in the in the lattice community so these are very natural conditions when you have a partial order then you might want a biggest element or a smaller and the smallest element and this for example first v would be the biggest because any other element in g of e is a subspace and the partial order is being a subspace of and the smallest element would be zero the the set zero because it's in every uh linear space distributivity is rare i won't dwell on this it's interesting and it uh leads to many um other notions but this is luxury distributive and uh the power set of a set x with inclusion is distributive as you can quickly uh figure out but the one we're interested in is not distributive as you also can quickly see um so um you we really need to get around this property and that's important so there are other partial borders which you can think about to be a multiple of and there is a notation i'd like to introduce which will come back later it's this a with a down uh an arrow pointed downwards which is the set of all elements smaller or equal to a and it will play an important role now ortho modulo lattice means a lattice with at least a free complement for the moment in the sense that for any a there is some complement but it might not be unique like a complement subspace to a sub vectors so sublinear space of v is not unique and so yes a plus b equals c means that the join is c and the meter zero zero you see was the smallest now author complemented and you see the analogy with orthogonal complement is much more restrictive you want uh that a and its or its complement have nothing to do with each other or little to do with each other mainly their meat is zero is the minimal element you want this compatibility with the partial order and you want involution and if you look at the details you'll see that this is already preparing for a real complement rigid complement as it's called in some parts of the literature because then you have a separation of the maximal element into a and psi of a and that's what we need because we want to separate holomorphic from polar okay so there's also a notion of relative complement which is important because you don't always work with the whole space you might work with subspaces and want to separate them into a sub uh another subspace smaller one and it's complement and so an example yes i won't go into those examples the the they're very uh they become they belong to folklore knowledge uh so taking the or ordinary complement map in a set and um and then taking the orthogonal complement which um gives you all these properties for when you have the luxury to have an inner product around to help you so any question maybe okay so this is uh preparing for us to see how what we need of a compliment um so locality now because remember we wanted to go from locality this symmetric uh binary form to a complement so let's see what what locality is it is indeed a symmetric binary relation so you might think oh it won't separate much if i take the identity for example which is symmetric binary but that's true and we will rule out that kind of uh relation very soon and the polar set i told you so here we're on um for the moment we're on the set and the polar set is all the ones that are have nothing to do with a that are independent we would read this
a independent of b so all the b's that are independent of a and what i called i could like to call polar set for various reasons it's like a duel and if you note a is is in its double polar set so recently uh we came across this uh statement so um yes which is in fact uh well known in the lattice literature but i was not at all aware of this literature so uh we were informed by the referees so um a locality on a post set is now we we're putting a poset structure so there was no partial order here there is and so we just have to make sure everything is compatible so the polar sets of two elements one smaller than the other are included in the reverse order in one and another this is called a galwa connection surprisingly for me absorbing this is saying that if something is orthogonal to the bigger element then it's orthogonal to or independent of the smallest one and then you see we had a was in this double polar now we're asking that all the elements below a are in the double order and all these conditions are equivalent and this is called we call it a locality poset but it's called weak degenerate orthogonal if and correct by some authors the the terminology is not uniform in that literature it seems so now we're putting a little more we're putting a lattice so now we have two operations and we have to make sure everything is compatible uh we have the partial order but now we have also the join and the meat for the meat it's automatic due to this partial order this is the smallest of the two for the partial order this is not automatic it's the largest of the two and uh you you see we start seeing this locality constraint well the fact that this five we had originally should um should uh um obey some locality or should be compatible with locality and this is what is kind of coming into the picture if a is independent of b1 and b2 it should be independent of their join you can view it in a different way it's equivalent saying it's a sub lattice or a lattice ideal so example uh this very nice but not really interesting example of post sets it's too nice um you can equip it with a non uh with a locality which spoils the postset locality if you take the instead of taking the intersection to be void yeah by the way this locality relation reflects what one might intuitively think of with locality if you think for example of smooth functions and say two smooth functions are independent if their supports are disjoint and this is very much used when you deal with divergences whereas here it's when their union is x and that's not compatible with the posted structure and of course the orthogonality we just check all along that we have it with us yes we have because that's what we're going to um to generalize and so now remember we wanted to separate um things because we wanted to separate holomorphic from polar and so now that's where we're introducing these uh conditions and here it is you see if a is independent of b then their meat yes their meat should be minimal uh you could see it as a non-degeneracy condition if a it's a non kind of a non-reflexivity in a sense this has another meaning but a should not if a is in relation is independent of itself then it's zero so this is the very important separating condition it a this locality relation is actually separating uh this is just asking that the polar set of zero is the whole thing and then we have this completeness so remember that we were at this stage we had a we started noticing that a was in the double polar because of the compatibility with the partial order we have that all elements below a are also in the double polar set now we're asking that they coincide this is equivalent to a being the maximum of this set and if you remember there was something about a maximum at some point when we looked at the relation between orthogonality and orthogonal complement this is preparing for that okay example it's always the same example just to make sure we have it with us yes everything's okay now we're going to the core of the uh our purpose namely remember i wanted to relate orthogonality as a bilinear relation sorry binary symmetric relation with the notion of orthogonal complement and we wanted to generalize this now so we realized thanks to a referee that this theorem in a very different language was already existent uh we proved uh corollary as well which was not there but essentially this is known and i don't think everybody in lattice theory is familiar with this reference because it doesn't seem to be quoted very much but this belongs to folklore knowledge we realized in lattice theory that an author complementation is in one-to-one relation with what we call that's not the terminology used there and it's not formulated in this way strongly separating locality relations so how do we do that so this uh this is like the orthogonal complement map and this is like the orthogonality and strongly separating remember um it's really uh making sure that things um uh combine into are separate so maybe i should make sure strongly separating was this and this is more stringent than this ordinary separation and this is going to ensure that we can go from ortho author complementation to a strongly separating locality relation so how does it go you take a symmetric binary relation t and you build your psi t psi t will be the max of the polar set remember that's how we build psi orthogonal the other way around how do you decide when you've got a complement map the two elements are independent if the one is inside the well i should have put this is not quite right i should have put or the other way around sorry this is a mistake
b is in um the complement of a or conversely so okay i'll skip this this is a refinement of what we did and uh what's important is that when we put here the orthogonality we have here the orthogonal complete complement we saw that when we discussed this example any question okay i have a quick question sorry yes um so the strict the strictness of this complementation is this required for the renormalization application in the linear poles yes yes because otherwise you don't have uniqueness you see you have as long as you don't have this strict requirement strictness requirement there's a there's a freedom of choice what complement you're going to take there's no map really you can't define a map saying this is going to be the complement yeah abhalidis and pomezheim when they looked at when they looked at polytopes and the euler magnarian formula polytubes they called it rigid complement it's a good word maybe rigid thank you pleasure so we were working on lattices and that served for us to be able to find you know all uh abstract locality relations which would serve a separating purpose but now we have to go back to the vector space and so let's go back to the vector space and we first equip it with a set locality but now we have linearity so we require that the polar set of any subset is a linear subspace if you think of what happens for the orthogonal complement the orthogonal complement of any subset is a subspace yeah so that's the kind of thing this is very it's a difficult condition to deal with in other aspects of this locality framework um okay so we shall say that this locality on the linear subspace is non-degenerate if we have you see it looks very much like what we had before but now remember we're on v we're not on spaces but this will imply that the polar set of v is zero and strongly non-degenerate means that we have it conversely if the polar set is zero then u has to be the whole space and so interestingly uh there is a one-to-one correspondence between these localities so this is looks very weak set locality which has some compatibility with linearity the orthogonal or the polar is linear this is in one-to-one correspondence with other ones we've been looking at lattice locality relations on this lattice g of v dov is somewhere on this lattice um and how do we set the relationship the the relation between the two two spaces are two vector spaces are independent if they're elements their vectors are independent this is here conversely um two vectors are independent if they're if projectively if they're the one-dimensional spaces spanned by these vectors are independent in this g of e and there is a one-to-one correspondence between what we've been looking at up to now so locality relations on this lattice and the ones on the vector space and if you want something more stringent like strong loc strong separating one the one that we're going to use indeed they are also in one-to-one correspondence that's why we went to lattices because that's where things are more understandable um so as a corollary since we know that uh this special class of locality relations is is in one to one correspondence with author complements on lattices it gives you something on vector spaces and that's a translation of what we had before that's what we aimed for because we needed complement author complementations on vector spaces and we get them through ortho complementations lattices and so all this is just translating putting back to the vector space what we've learned on the lattice and it generalizes uh what we knew before namely going from an orthogonal complement orthogonal relation orthogonality relation to an orthogonal complement now why have done all this work i think i have two more minutes uh you might say well most of it is maybe orthogonal complements but i don't think so and this example which i don't think i have time to comment too much gives you a way of building from in fact it's like building from a set complement of vector space complement if you analyze the way it's done there's a separation of a set of angles into two disjoint sets of angles and the bijection which takes one set to the other and um then so u theta which is there it's a line in the direction theta is sent to u psi theta psi being uh an author a complement in fact on sets in a sense okay and then uh if you take psi to be pi minus psi of theta to be pi minus theta you get back the orthogonal complement this is one way on r2 to get other complement maps which are not uh uh ortho complementations which are not the orthogonality now why do what we want to do as from there so this was like a sidewalk to understand all the ways we have separating uh on the vector space now what we're going to what we're doing now is um generalizing this lauren expansion construction beyond the orthogonality relation and for that we use the language of cones we translate um this this these laura expansions into a conical language which makes it much more attractive now we also looking into the galway group and that's why we want to go you see we don't really need lauren expansions to have a minimal substraction scheme because we don't need to look into what i call the dustbin at some point which contains all the polar parts but we do if we want to look in a more refined manner manner at all transformations linear transformations of multiple variable metamorphic germs with linear poles which stabilize holomorphic germs and then we really need to look at what's in the dustbin namely at the details of the loja expansion and we want to do that for this locality but also beyond and we hope that this can give us some insight on what in this picture could be something like a renormalization kind of group so thank you for your attention and here are some um references um and thank you very much all right thank you very much sylvie are there questions
well we're waiting for other people i have a question so how important is the boundedness it comes up in the definition of the
compliment uh but you know you could have a locally complemented yes that's right thank you for this
question we indeed a relative complement is is enough but you can always when you um so you can always consider when you have a lattice which is not bounded the lattice of elements smaller than a certain element and then this one is bounded and you so you can always reduce your your study to um to the case of a bounded lattice and this is called relative complement the one you mentioned and indeed for example as i mentioned in uh well everywhere in the co-product for feinman graphs it's a relative complement because you're taking as a graph and you're taking a sub graph and a complement with respect to the bigger graph the same for trees it's a sub it's a relative complement and the same for gahufalidis and promescymen polytope so usually you do need a relative complement but you can always thanks to that reduce your your investigations to a bounded lattice i see uh david broadhurst has a question
thank you silvia for this very clear account there was something that
intrigued me at an early stage where you pointed out that there was a sort of ambiguity in what you were doing by taking partial fractions of your l's yeah oh yes when i had z1
minus z2 over that one plus z2 no no no
no no one upon l1 one upon l2 i can express as a sum of two terms using different
partial fractions
okay maybe i'll go back to it ah yes
yes i know no no the logarithm expansions yes here so so so can you carry that ambiguity through your work i mean yeah yeah that's the point uh so uh it uh thank you for this question so what we're doing is not unrelated to blow up procedures which i'm not at all competent on but uh when you kind of split the divergence into you kind of uh blow it up to understand how it's built and this is what we do in introducing multi variable regularization schemes or germs or functions and we keep track of so when you do a blow-up you look at things locally we do it globally but at the cost of having two things a an inner product which is my locality which rigidifies so we're in the we're in the category of set with locality and the second thing as you rightly point out we have to keep if we want to the details of this expansion if we really want to know what happens in the dust bin what i call the dustbin and not only do a minimal subtraction scheme for a minimal subtraction scheme we don't need to know what happens here if we really want to know that then we have to look at this family and what i pointed at here is that it can we have to fix it if we want to a certain laura expansion and this is kind of replacing the other option which would be to look at things locally setting giving ourselves a family of supporting cones is a looking glass into this uh the the singularities of f thank you does that answer your question i have a maybe a related question yes on that slide so so when i think of regularizing things with with many variables then there's many ways of doing it is it the case that the queue like the dependence on the locality this is how you encode making the choices of how to do the partial fractions that's right and that's why you see that's exactly the point q it determines uh who is polar because h j should be kind of orthogonal to the allies uh in a metaphorical manner so yes and it uh it also so this decides who is polar and then it all goes in there and that's why there's a plus with a q because the here it doesn't show but we've cho we've used the inner product q in an essential manner exactly and that's the choice of q is a choice of renormalization in a sense and what we want to do is you know go beyond q because we feel the importance of q and that's why we wanted first to make sure we understood the role of q and that's what we these uh stringent locality structure sure but uh but within this very special case um if i change q do we have like a renormalization group formula that's right it's part of it it's part of it okay that's what we're doing you see and our first attempt is this uh galwa group uh so the gagwa group will look into that so before we go to a more overarching uh renormalization group we're looking at the galway group of transformations of such metamorphic germs which leave this part invariant i mean each argue each individual invariant which is identity here okay and this relates to transformations of cones because these are described by cones yes so yes that's why we're looking at this catwalk group exactly because we know we feel it's obvious that the renormalization group has something to do with this choice yes thank you very much thank you so sylvie there's also a question in the q a so jonathan in the q a asks about metroids he says the concept of metroid generalized linear independence and he wonders does your theory also apply to matroids i've never looked into that i did look at some points that matrix but not in this context i don't know so i suppose he had something you have something behind your mind to ask this because you think it could apply maybe i'd be grateful for a reference so that i could look into it no i haven't thought of that eric do you want to unmute jonathan he's one of the attendees uh i don't seem to have that power today but but eric should then you can
tell us uh more about your question if you'd like yeah jonathan should be speaking now oh
he says yes
okay okay well thank you jonathan
thank you maybe you can send me a reference
thank you are there any other questions
all right why don't we all thank sylvie again he can unmute and clap thank you very much thank you