 in terms of classifying semi-simple cumulological field theories in terms of a topological field theory in this R matrix action. So I want to first define given tools R matrix action on logical field theories and then use that to actually write down a family of topological relations on MGN bar extending the Faber-Zogger relations. Start with given tools R matrix action. So the situation here is that we start with a logical field theory, omega. And remember, there's various data that this consists of. You have classes omega GN of a vector space V. You have a metric on the vector space. You have a unit element. It gives rise to more structure such as the quantum multiplication on V. If we are given such a cumulological field theory and then a matrix R, so R should be should have leading term one and then it's a power series in some formal variable Z over the vector space V. Possibly to be careful, I should be extending things to complex numbers or something, but power series in Z. This matrix R has satisfied single conditions in plectic condition. So times R star of minus Z should be equal to one. Identity matrix where R star is adjoint with respect to the metric. So that's the starting data and then given those R matrix action should be a new cumulogical field theory. Comulogical field theory. And that means a class for every GNN and said cumulogy of MGN bar tensored with N copies of the same vector space. The other data will be the same. All right, so the way in which this will be defined is a graph sum. Tell you what you get when you pick GNN and then pick N vectors V1 through VN. To feed to it, to get an actual cumulogy class in MGN bar. A big sum over gamma stable graph, one over automorphism group of gamma, iota gamma star. Remember this is the gluing map corresponding to gamma gluing together a bunch of MGVNV bars. But I now need to tell you what to put inside here. A product of our vertices, but then for each vertex, I want to take a sum over non-negative integers k sub V by factorial of k sub V. So k sub V is going to be a number of sort of virtual marked points which are added in. So what I'm going to write here is pi star, push forward along pi of something where pi here, I'm just using as notation for some forgetful map, which is going to be from MGVNV plus kV bar to M bar GVNV. We're getting about the last k sub V points. So we choose for each vertex, we choose some number of additional marked points. I'm going to tell you a class on MGVNV plus kV bar and then going to push forward and divide by kV factorial. And that's going to be the contribution vertex V. You then piece those all together using the data of the stable graph gamma. But what I put here, I should at some point be making use of both pieces of data here, the homological field theory and the R matrix. So here I'm going to put, maybe I'll just write it for now as omega GVNV plus kV. Then I need to tell you some vectors to feed to it. I'll do that separately in a picture. This is the shape of the formula for Given-Toll's R matrix action. To sum over graphs where at each vertex you essentially put the homological field theory to start with. So R doesn't appear here and I also haven't told you exactly what NV plus kV vectors you feed to omega. So the picture, got enough room. Let's think about this in the case where over here I'll be thinking about the case where gamma is very simple graph. Say it has two vertices, a single edge. I haven't told you how R interacts. In the case where R is, yeah, you will be able to observe that from the formula. More or less what will happen is that there will be edge terms, there will be edge factors, vertex factors and leg factors and if R is one then the edge factors will vanish. Which means if you have one edge in your graph gamma, then the contribution will vanish and you'll only get contribution from the stable graph with no edges. The one corresponding to smooth curves. Okay, so I want to talk about some example, say a very simple graph. I have two vertices, maybe I have leg one here and leg two here. So that's going to be the gamma, what I've drawn so far. And then in order to describe a term here I also need to choose some number of virtual legs to add to this graph for each k. So let's say that, I don't know, maybe I have two legs here to draw by dotted lines, one leg here. So kV is two there, kV is one there. And these virtual legs, they don't have labels on them or anything. I mean I could write like labels and V plus one and V plus two on here and V plus one on this leg. But now I want to, so this is the basic situation. I have over here, this vertex corresponds to some gV and V plus kV. In this case, nV is two and kV is also two. So now I need to write insertions everywhere. So let's start by on the legs that we start with. So here we want to at some point incorporate the data of the V one through V N that we started with. So going to do so just by writing, let's write it over here, R inverse evaluated at psi times V one. Contribution on that leg. Similarly over here, we'll have R inverse of psi V two, label two, label one. So the way you should think about this is that V one is an element of the vector space V if you're a homological field theory. And R inverse of psi is some endomorphism of V with coefficients which are power series in psi. And multiplying then gives some element of V tensor, the element of the tensor formal power series in psi. So the thing that I'm putting here is really, should think about this as being in say Q, let's just say in V tensor formal power series in psi. And that's going to be one of the factors that I put in here. And you might ask, why are there psi's showing up here? And the idea is I'm extending the homological field theory sort of linearly in powers of psi in some sense. Yeah, so here I'm at a single leg, so I take that psi there. I could give it a name if I wanted, but I'll have psi's appearing everywhere. Basically this formal variable Z will be replaced by psi and I just have to tell you where to use in R and I just have to tell you where to insert R. So at each half edge in this picture, both these five legs here and the two sides of this leg, I'm going to tell you some vector in the space of V weighted by power series in psi. Then I feed all of those in here. I apply the homological field theory, I get a class and then multiply it by the remaining powers of psi. Get something which I then apply these other operations to. Okay, so that's what the actual legs are. Then there are these sort of virtual legs, capo legs sometimes they're called. So there I want to put something which I don't have any visa buys to use. That at each one of these I want to put something like psi times one minus R inverse of psi. That's some matrix applied to the vector, well the unit vector. And I put the same thing over here. I'm using a different psi on that leg of course. And the same thing on the third leg. Then along the edge, this is the most complicated part. Along the edge I want to come up with a bi-vector so that I can feed one element of vector space V to the left and one element to the right. So I'm going to write a bi-vector here, previously using this bi-vector which is the inverse of the metric, A to inverse. I can take that and subtract off, take our inverse of psi acting on the left side of Aeta and also acting on the right side of Aeta. Principles there, acting on the right side of the bi-vector. Again, A to inverse is in V tensor V. Take that and then I divide by, sorry this should be psi prime, divide by psi plus psi prime. So here I have two psi classes, one for each side of the edge. One psi class really lives in, in this case I have like MGV comma four on that vertex. Other one is an MGV prime comma three. I have the psi classes along this edge and both sides of psi and psi prime. You might be a little concerned I'm dividing by psi plus psi prime here. It turns out that the visibility of the numerator by the denominator just as formal series in I guess V tensor V tensor power series in psi and psi prime. This formal divisibility is essentially equivalent to the symplectic property or at least it's implied. Sorry, there's no star but there's a transpose here. I mean this is a little different because I'm really I'm acting on just, transpose means I'm acting on the right V instead of the left V. I'm acting in the same way. I mean I think it will be equivalent to what you want to do and it's just a different notation. This is maybe a little clearer what this means but comparing it with what's over there might be harder, yes. Okay, so I've now told you factors to put along every half edge, which again will in general be vectors in my vector space weighted by power series in whatever psi is at that location. And then at the vertices I feed these inputs to the omega GV and V plus KV. This is all the stuff going on in the definition of the R matrix action. I mean more or less you have, and there's this weird thing going on with these virtual legs which I'm then removing by the push forward map by this one over K factorial pi star where pi is forgetting about those K those are the dotted legs here. The purpose of that is actually, so you might notice that that's the one place where the unit shows up in this definition. The insertions that I do along the virtual legs and that's because the purpose of the virtual legs is basically to preserve the unit. You can define the R matrix action without the virtual legs and it won't preserve the unit and it will preserve all the other structures. For the rest of what's going on, these are more or less the most natural choices for inserting vectors that you can come up with if you want to twist by the matrix R. I mean, so now if we think about what happens when R is just one, the identity, this should be the trivial action. Indeed R doesn't change V one, so you just have V one applying there. These virtual leg contributions are zero, so you don't have to worry about them. You can assume that KV is zero everywhere because these virtual contributions vanish and the edge vanishes also into A to N first minus A to N first on the numerator. So that at least should convince you that and so you're left with just a sum over a single graph. No edges corresponding to the generic boundary stratum I guess. And in that you have one term which is just omega GN itself evaluated at V one through VN. So that should at least convince you that the action of the identity matrix is trivial. The fact that this is a group action is quite a lot to check. I mean, the fact, and there are many things that could be checked here. You can check that this takes comological field theories to comological field theories. And you can check that this is actually, so that requires checking the three axioms I wrote down previously. And you can also check that this is a group action of this group of matrices, sometimes called the symplectic loop group. And I should note that again in this formula R itself doesn't appear, only R inverse appears which might look a little silly and that's just so it'll be the left action so the right action, the usual thing. So that's totally for that purpose. Okay, so this is a long definition but I mean the result in practice is that if you have an explicit comological field theory that you start with and you want to compute the action on it by a specific R matrix, then you end up with some sum of graphs. And the various factors that you put in here, you get certain psi factors coming from the matrix R and although it might be less clear, these virtual legs are more or less contributing kappas. So you're getting sort of arbitrary topological classes showing up. I should mention that in this definition I'm only using like gluing maps and psi classes and push forward maps by forgetful. I'm using all these topological construction so it should be clear from this definition that if I start with a comological field theory which takes values and topological co-homology then I apply some R matrix to it. It will still be topological. Any questions about R matrix action? So maybe I will restate now the theorem that I said at the end last time Telemann of Given-Toll's reconstruction conjecture. We previously said that, wrote it as in the form of any comological field theory as the R matrix action acts freely and transitively on the set of comological field theories with a fixed V eta one and quantum multiplication on V. Maybe I'll state it in a equivalent way. Is that any semi-simple comological field theory omega can be uniquely written in the form omega equals R times lowercase omega R applied to lowercase omega. For some R in the symplectic loop group, I guess depending on your definitions might be only part of the symplectic loop group, this group of matrices, and some topological field theory lowercase omega. Remember topological field theory is just a comological field theory which only takes values and comological degree one. So you just have numbers. This is equivalent to the same end I gave yesterday. I think I briefly said that it was equivalent to this. It means this symplectic condition here on power series. What? The R of Z times R. Yeah, this. All I know is that sometimes people call this condition saying that R is in the symplectic loop group and sometimes it's that R is in some piece of the symplectic loop group. Maybe I shouldn't use the terminology if I'm not sure exactly what the symplectic loop group is, but I mean that condition. Where do I need it? Yeah, so as I mentioned out loud, you should worry a bit that I'm inverting psi plus psi prime here because if I want to end up with power series and psi and psi prime to feed to the two sides, I shouldn't be dividing my psi plus psi prime. And the symplectic condition is basically equivalent to the statement that the numerator is divisible by the denominator here. Yes, I mean it's going to be symmetric again by how I'm thinking about this transpose. Like eta inverses some by vector which is symmetric by definition because I started with a symmetric by linear form. Eta inverses symmetric element of v tensor v. And then I'm really just, this is some element of endomorphisms of v. I'm applying it to either side, so we'll stay symmetric. Sorry, this is psi prime here, maybe that's not. Yeah, okay, so I mean, when I write this down here, I mean the first factor v is the left, the right side. First side is psi and the second side is psi prime. All right, so by this theorem of Telemann, we know that any semi-symbolic homological field theory really is of this form and so of this form where this inner homological field theory is not an extra homology class. Like in general, if you apply, given those are matrix actions some homological field theory and you have the sum over graphs, but then you also have the sort of arbitrary tautological classes on the inside in this omega. But in the case where you're playing our matrix action two topological field theory, this is just a number and in that case, you really have that this is an expression of the type of talk to help before for a tautological class, writing it as a sum of basic classes. Push forward of under gluing map corresponding some graph and in the case where omega is just topological field theory, this entire thing in here will just be a polynomial in the kappa classes and psi classes. This is a powerful theorem that basically says that, again, if you have some tautological class you're interested in, if it belongs to homological field theory, then there's a sum R matrix which will give you a formula for it of this form as a sum over graphs written explicitly as a sum of basic tautological classes. These additive generators for the tautological right. In particular, and I say this out loud last time but corollary, maybe I'll just say it out loud again because I don't really need it, but corollary is a semi-simple homological field theory takes tautological values, at least in homology. So let's give in those matrix action. Now I want to move back towards relation. So I'm going to write down a, I'm going to describe to you a homological field theory in this form. I'm going to tell you a topological field theory. I'm going to tell you an R matrix. Then by using this action that will determine some more complicated homological field theory given by R times lowercase omega. So by telling you in the description of a specific topological field theory, I'll just call it lowercase omega. That's why I've been calling all my topological field theories. So to tell you what this is, I need to tell you first all the basic data of what I should have a vector space. The vector space will be two-dimensional. Let's give it a basis. It's Q E zero plus Q E one. Two generators. They need a metric, data, be the anti-diagonal metric in this basis. They need a specific element, the unit element that will be zero, the first basis element. Then I need to tell you what omega GN of arbitrary inputs are. I'm going to, because again, homological field theories are symmetric, I just have to tell you what it is evaluated at E zero with N zero copies, tensor E one with N one copies. So N zero plus N one equals N. I want this to be equal to either two to the G if G plus N one is odd. Again, I'm just, I have a number for every GNN makes it topological field theory. Don't need to talk about comology. Or to check, this is a topological field theory rather check it's a comological field theory. I would need to check those three axioms. Well, it's symmetric by definition. Symmetric on this basis. I just care about how many E zeros and how many E ones I have. And so I just need to check that the, it behaves well under pullback by gluing and forgetful maps. And this is straightforward to do. Basically what's happening is, well first of all pulling back by forgetful maps should be the same as putting an extra copy of E zero N. And however many copies of E zero have doesn't affect the value here. Of course, pulling back by any of these maps and these are just numbers. Pullback doesn't change the degree of a number. So pulling back by forgetful maps is fine because E zero is the unit. Pulling back by gluing maps is a little bit more interesting. Basically what's going on is that you start out with some E zeros, E ones. Then you want to pull back to some boundary divisor. And so you have still your N, E zero, and E ones on the two sides here, zero, zero, one, something like that. You have some genus here, one, G two. And that's just a data of which gluing map I've chosen. And these are the original E zero and E one inputs that I had before taking a pullback by this gluing map. And now remember that the way that pulling back is supposed to work is that I insert the bi-vector given by the inverse of that. So that bi-vector means I either put zero one or I put one zero out of those two options. I put something different on the two sides because I have the anti-diagonal metric. And then I'm supposed to sum up those two values of omega GN up here times omega GN up there for each of the two options. I'm supposed to get omega GN of the thing I started with. And the reason why this works is that G one plus G two equals G, and exactly one of these choices will have this parity constraint be satisfied. There's something similar for the self note, okay. G minus one, except here, the situation will be that the genus went down by one, but both possible ways, either zero one or one zero will maintain the parity constraint. That's just very, very quickly explaining what you need to check that this is a homological field theory. Yeah, I mean, more or less, what you want to do is you want to choose some, and basically you want to change basis in V to some orthonormal basis for the, for some basis that works well with the quantum multiplication. And once you do that, the formula becomes light. There's some general role that one can write down. I mean, one can classify topological field theories in terms of, I mean, you have to tell me what all of these things are, but, it's not even a collection of matrices. It's a collection of numbers indexed by tuples. Yeah, it is not a hard check. Okay, so I want to take that topological field theory. The dimensional vector space. And I'm also going to take an R matrix. We'll take number R should be a two by two matrix now with entries which are power series in a single variable Z. I'm going to write it in terms of these hypergeometric series A and B. Of course, I'll leave the picture up there, just really the content of how our matrix action works, but right up here. So remember that we had, in defining the FC relations, we had hypergeometric series A equals summation six n factorial for three n factorial, two n factorial, t to the n, and B is something similar. I want to write down a two by two matrix R now using those. Rather using the odd and even parts of A and B. So, make sure I have the signs, okay. Negative even of Z, odd of Z, negative A odd of Z, A even. And even, and odd just mean I take the even powers of t or the odd powers of t, and then I replace t by, what I'm choosing to call my formal variable here, Z. So this is a matrix. You can check that it has leading term identity matrix because A, I mean, clearly has diagonal leading term because in Z, because odd parts means you don't have constant terms off the diagonal. A had leading term one, B had leading term minus one as I defined it. So this is actually going to be as a matrix valued, as a power series that's matrix coefficients, the constant coefficient is identity matrix as it should be. And then the symplectic condition is, you can check it's equivalent to a sort of nice little identity between the A and B series. And then I'll actually write what B is. 6n plus one divided by 6n minus one, 6n factorial, 3n factorial, 2n factorial. Symplectic condition, if you multiply this by its adjoint with respect to anti-diagonal, then you get the addition you need is equivalent to A of t times B of minus t plus A of minus t plus B of t equals negative two times here, of course. That's a somewhat fun identity between hypergeometric series to prove. And I really mean the symplectic condition is equivalent to this four matrices of this form. Yes. Okay, so I've now told you an R matrix and I've told you a topological field theory. I now have the R matrix action. It defines a more complicated comological field theory than this topological field theory. So now, if you look at given tolls or matrix action, think about applying this to it. I mean, it's obviously complicated exactly what's going on, but you end up with some sum over stable graphs where for each graph, you take a product of lots of local factors involving taking these A and B series and inserting them in various places as powers of psi classes and stuff. Should start to look similar to the FZ relations I wrote down. Work of punterbonda, so, I should say that our proof worked in comology. Later Felix Yanda proved this in Chal. Again, you can have Chal ring value theories if you want and define the R matrix action the same way. Applying this R matrix, this topological field theory defines a Chal value theory if you want. What does the theorem say? As if I take that R matrix and that topological field theory, mine them together, get a comological field theory, evaluate it some more, let's say, that N0 copies of E0 and N1 copies of E1, then the theorem is that this vanishes in degree, comological degree, D vanishes in each star of MGN bar, D for all D strictly greater than G minus one plus N sub one divided by three. Pick G, pick N, zero and N1 summing to N, pick D greater than this number, then take this data together, combine it using this picture here to get some overall stable graphs of these topological classes given by these series, take the degree D part and the thing that you get a topological relation. Yes, I will briefly sketch the proof. Yes. So I'll just write in parentheses. So this gives non-trivial topological relations because the thing that these vanish and they certainly don't formally vanish if you write this out. And this is slightly a lie because if you look very closely at this, you can see that there might be some sort of parity constraint lurking here, which again shouldn't be surprised from the FC relation since you have this anti-diagonal matrix. Your matrix R has this even and odd thing going on. Combine them together, you still have some parity thing which means that actually half of these vanish formally without giving relations. But the other half give non-trivial topological relations. Okay, so let's sketch the proof. There's the geometric input and that's the, just say, the existence of and properties of Whitton's three-spin class. Rather technical to define, it was first defined by Polish talk. Later by Kyoto and various other people, Van Jarvis were on analytically, for instance. And so what do I need about this class? I need that after a shifted version of it, a shifted version, the three-spin class forms a homological field theory, we'll call W3 the following properties. The very special homological field theory. So properties are that W3 has degree zero part equal to the omega that I previously defined, that topological field theory. W3GN, if I evaluate this again at E0 to the N0 times E1 to the N1. And this should be equal to, all of these things are up to some scalar constant factors, by the way. I think for this to be correct, I have to multiply by 1728 to the N or something like that, given how I define things. But I guess this part will be fine. But Whitton's three-spin class, which is of your homological degree equal to the number that I wrote over there, G minus one plus N1 divided by three plus lower degree, lower degree terms are due to this shift that I haven't told you about. But it's in the top degree pieces of this pure degree. It's equal to when three-spin class above there, this homological field theory vanishes. Now from this theorem, you can guess what the final property is that I made, which is that in fact, W3 is equal to, up to some factors of 1728 maybe, equal to R applied to omega, but R and omega of that theorem. So how do we know this? So this is where the A and B series actually come up. We know this because of the second part of theorem, I certainly didn't write down, which tells under certain circumstances how to explicitly determine the R matrix. And so here this comes from R matrix is obtained by solving a fairly simple, I'm not going to write it down here, differential equation coming from there are a couple of equivalent things I could say, coming from some additional structure on this topological field theory in Euler field. It comes from homogeneous calibration of the corresponding Frobenius manifold. I don't want to get bogged down the details, but the idea is just there is some additional sort of numerical invariance of W3 spin class related to the fact that it was a pure comological degree before I shifted it, which give rise to some straightforward differential equation that lets you compute the R matrix recursively degree by degree. And when you do that, you get these six n factorial by three and factorial two n factorial. Really in this case, the A and the B series are seen as basic invariance of some Frobenius manifold corresponding to this three spin class. So yeah, so you can do this in general if you replace three by R. So there is a differential equation that it always satisfies. However, that differential equation doesn't have unique solutions. If you also additionally have some Euler field or homogeneity condition, then there is in general differential equation that determines R uniquely. In practice, if you want to know the R matrix, you don't always need an Euler field or homogeneity constraint, but in this case, in this case, that's the appropriate method to use to determine the R matrix. The theorem is that there's always unique R matrix determining that R matrix might be hard in general, but in this case, there's some differential equation that we have. And yeah, there's some such equation in general. All right, so this gives a lot of relations. I've told you that they are related to the FC relations. You might wonder about exactly how they're related. This G over three looks familiar, but the FC relations had this parameter which was a partition, which isn't really visible here. So that concludes the sketch of the proof and you just combine the fact that this geological field, Terry vanishes above a certain dimension by geometric reasons with the fact that we have an explicit formula for it to get these relations. We'll erase this now. So let's R of G and D relation R star of MgN bar, really it's an Rd of MgN bar given by degree D part of what is written over there where I'm just using E1. So take R applied to omega GN, evaluated at E1 and copies of that and takes the degree D part. I'll state a conjecture about the full structure of teleological ring of MgN bar into two ways. The first way, the conjecture, all relations in R star of MgN bar can be obtained, the RgN and pullback slash push forward by and forgetful maps. What I mean by that is on MgN bar you've all these maps between the spaces. If you have a teleological relation in one location then you can take the pullback of them or forgetful maps, that's a new one, maybe multiply by some side classes and it's still a teleological relation. You push it forward then you glue it to some arbitrary other classes still a teleological relation. You've all these transformations that you can do to take teleological relations, teleological relations. This conjecture is saying that if you take all such ways of moving this small list of relations around, again there's some constraint here which is that D should be greater than G minus one plus N over three or the start to give a relation. If you take all of those relations move them around by the basic maps and you multiply them some things all possible ways and this is saying that the conjecture is that this generates all the relations in teleological ring time GN bar. Let me now write that a little bit more explicitly which should start to make it clear what the relationship is with the FZ relations in terms of where does that partition show up in all this. It'll be very explicit about which of those push forward pullback maps you need to use. The conjecture and it's a proposition, these two conjectures are equivalent. Again, I want to say write it as any relation in R star MGN bar is a linear combination of relations the following form. So I start with one of these RGND relations. I then pull it back by a forgetful map. Pulling back this forgetful map you might remember from conological field dairy axioms the same thing as adding inputs which are the unit. The unit was E zero. So this is just adding back in the E zeros which I left out in this definition. All this I don't know, pi two, pi, there also be a pi one. Now I want to take that and multiply it by a pi monomial filled teleological relation. Now I want to take all of that and I want to push it forward by some pi one star. Yeah, so pi one, pi two are forgetful maps maybe forgetting multiple points. So for instance, pi two is from, and these are all going to, pi one, pi two will both be from MGK to MGK minus, K prime or something like that. They're both forgetful maps of that form. So I've pulled back my forgetful map multiplied by some, I'll say, psi monomial. Now I push forward by some other set of points. Now I want to take all of this. I'm going to take the push forward by a arbitrary gluing map where this is on one factor and then the other factors I'm pushing forward corresponding to other vertices of gamma are just arbitrary by kappa monomials. Iota gamma is a gluing map. And these are clearly operations I can do in still the teleological relation at the end. I'm saying that these are all operations I need to do conjecturally. And if you don't do this final pushing forward to a boundary stratum, and instead, so then you leave out that part of this construction, then you restrict the interior to MGN, then you get precisely the FC relations generalized, as I explained before, to MGN. And the way in which the partitions come up is by exactly which monomial in the psi class you're multiplying by and which points you're adding and forgetting. Okay, I am out of time, so I will stop there. This conjecturally gives description to a logical ring of MGN bar as with the FC relations on MG in small cases we can check that these are all such relations because they give a Gorenstein ring. This will always be the case as if we know this ring is never a Gorenstein, but these are still all the relations that we know how to construct. All the special relations that have been constructed in the past like Getzers relation, R2M14 bar special cases here where you choose appropriate GND and maybe move it around a little.