 So yesterday I described these FZ relations as explicit list of all the relations we know how to prove on MG and sort of speculated these are all the relations on MG. And today I want to move back to stable curves and marked points. So I want to begin by sort of outlining what you want to do if you want to generalize these relations to MGN bar. So there's sort of two steps to this and I'm not going to, I'm not going to fully write out the formula today, but I'll at least describe approximately what it looks like. But the first step, generalizing FC relations MGN bar, the first step is adding in the marked points. So I mentioned briefly before that the tautological ring of MGN, like all the tautological rings for me is defined by restriction from MGN bar. And if you're just dealing with smooth curves, this means that you don't have any graphs to worry about. You just have kappa classes and psi classes. So this is a ring generated by one through psi n and the kappa classes that we had before. It turns out that if you want to generalize the FC relations by adding in those marked points, then you sort of do it in the, one of the simplest ways that you could hope for. So you have to change what relations are now going to be parameterized by relations, let's say relations in RD of MGN are, or these relations I should say, although again these are all the relations we know of, they're parameterized by, so previously we had a partition sigma, no parts that are 2 mod 3. We still have that, but now we need more data dealing with the N marked points along with non-negative integers a1, a2, through an, which are also not 2 mod 3. Note that they can be 0 though, and there should again be some inequalities, meaning the only finitely many relations, so previously the inequalities that I had were written on the board were that the bias of the partition sigma is less than or equal to 3d minus g minus 1, and concurrent mod 2. I need to modify that by adding in the a sub i's, so I add in the a sub i's. So there again only finitely many, for a given dg and n, they're a finite number of partitions along with a sub i's, and then what the relations themselves look like is going to be very close to the relations that I wrote down yesterday with n equals 0. For this it will be convenient in this notation, which maybe I should have introduced yesterday, which is that I define, so previously I had series a and b, these hypergeometric series in a single variable t. Now I want to define c sub 3i of t going to be t to the i times a of t, t sub 3i plus 1 of t is p to the i b of t. And as usual with these definitions, numbers are 2 mod 3 don't exist today. This is some sequence of power series. And the fc relations, as I described them yesterday, what did I do? I took the exponential of, I had 1 minus a, maybe I'll write it as c0 here, c0 is a, I inserted kappa's, took the exponential, I then multiplied by b, with kappa's insert raised to the power b is c1, power raised to sigma 1, c3, sigma 3. To get the idea, this is the same, this makes this formula look simpler if I have that extra notation. And then, this is all just the same formula I wrote then before with the c notation. I add this upper twiddley thing, changing the kappa polynomial everywhere and there, and then I took coefficient of t to the d at the end. I want to do the same thing here, except I need to insert the psi classes. Previously I just had kappa classes, one polynomial in the kappa classes, now I better have psi classes appearing in at least some of my relations as well. And I do so by just, at the end, multiplying by, I have ca1 of psi 1 times t, time, reach point, can, psi n of t. And then I still take the degree d part, the variable capital t, and this is just keeping track of the comb logical degree. So, this should be, right, equals zero since this is a theorem in rd of mgn. So, again, this is about as simple as you could hope for, for inserting psi's in there. You keep the formula the same, you just multiply by some power series in psi before taking the coefficient, before taking the degree d part at the end. And they're parameterized not by partitions, but by partitions with some slight decorations if you want, these additional integers. And the way you might guess this sort of formula is, for instance, when n equals 1, you're getting relations in mg1. You should be able to take the push forward by forgetting about the marked point. Forgetting about the marked point would turn the ca1 of psi1 t into some sort of expression involving kappas. So we turn into something very close to one of these factors. So the condition that the push forward of a relation should be a relation is more or less how you'd come up with this formula. I mean, it isn't logically implied by the regular fc relations. It serves the natural generalization. That's the first step if you want to guess what the relation should look like an mgn bar. Since the point is that an mgn bar, even if you just want mg bar, you'd want marked points because mg bar is all these boundary strata where you glued points together. So that's just an easy to state modification of the formula with psi classes. And to outline what the relations on mgn bar will then look like. So we have to remember what the tautological classes on mgn bar look like. They look like sums of these basic classes given by a graph and then a polynomial or really polynomial in the kappas and psi classes being pushed forward. So it turns out that these, so now we're into step two, which is also going to be the final step at least in terms of telling you approximately what the relations look like. So step two, these relations on mgn, now we, before we were generalizing by adding in marked points, now, now we're just extending by adding in boundary terms. So these relations on mgn extend to relations on mgn bar parameterized by the same data, which I've erased exactly what it was, but they were given by sigma and the a sub i with these conditions. The resulting relation, so if I write like r for relation of, then we have various input data, we have g, we have n, number of marked points, we have a homological degree, we have sigma, the partition, then we have a1 through an. That's the data we need to describe one of these relations over there. And the relation on mgn bar is given by the same data and it should look like equals, well it's the same, it's the relation on mgn plus classes supported on boundary, where by that I mean that the graph that you use should be non-trivial for those terms, should have at least one edge, correspond to having at least one node in your curve, and that's approximately what it should look like. Again, it's taking what you had before and extending it by adding something in the boundary, then if you take this relation, you restrict to the interior, you will get precisely that. Now the actual formula for it is going to look something like, I'm not going to carefully write down all the details right now, because we'll see another way of thinking about these formulas in a bit. But it will be a sum over gamma, a stable graph, okay. So now we want to write down some classes on this stratum here. It's really given by iota gamma lower star of some expression. Maybe I'll draw the picture here. So I have some graph, stable graph as vertices connected in some way, maybe we have a loop, we have a couple of legs, two, three, each of these should have a genus, which I'm not going to write down right now. This is what the graph looks like, and I need to tell you where to put cap as in size, yes? They're vertices. I'm writing them as loops because I'm going to write stuff inside them. Possibly I should have made the circles even larger, but that would have made it even more confusing between them and loops. So I need to tell you how to insert cap and psi classes here, decorate this graph with cap and psi classes. And the basic idea is you put an FC relation on each vertex. This isn't exactly right. So I can say a little bit about, but morally this is what's going on here. So by FC relation here I mean really this MGN relation because I have marked points. So each of these vertices has a genus, it has some path edges coming out of it, it corresponds to some MGINI bar, and I take some FC relation parameterized by some amount of data in that location. That gives me some cap as in psi classes by this formula. I insert it in each of these vertices that I push forward by the graph. Right, so there are some details which I am going to gloss over a bit right now because we'll see them a bit later, but you also have to choose a way of dividing the partition sigma among the vertices of the graph. Sigma equals like this joint union sigma v, v in the vertices of gamma, and you also have to choose what all the a sub i's should be. So this tells you what the sigma should be at each vertex. Just summing over all the ways of assigning sigma to each vertex, I also need the a sub i's. And it's clear what a sub i's I should use on the legs of the graph. I still have the a2 that I started with, the a3 that I started with, a1 here. But on the edges there's something more complicated that has to be done. And there is some sense in which what's going on is I also have to choose numbers which should be minus 2 minus x along each edge here. This is not the way we'll see it stated later, but morally what's going on is I also have to choose for each edge, I choose xe and minus 2 minus e. And then these are the other inputs I use for the a's, so also choosing these xe's. And what ends up happening is that, this is something that you should have asked about when I first said I was putting an FC relation on each individual vertex. If I put an actual FC relation on a vertex, forget about boundary issues, I should expect that that's already zero because it's a relation. The things I'm putting here are going to in general be FC relations, but they're given by this formula, but they are in the, remember I had this inequality relating g, d, sigma, and the a sub i's. They'll be in the range where that's, they might be in the range where it's not a relation. So use the same formula to assign cap and psi classes to each vertex, but you aren't just putting actual relations on each, you're just using the same formula. Okay, so this is just the sketch of what the relations look like. Again, I'll tell you specifically what the relation mg and bar is later. The idea is the principal term is when your graph just has a single vertex, no edges, and then you just have these terms here. What did you say? Yeah, so you have to, and that's something else. We're here, we took the coefficient of t to the d for each term individually, and we also can think of it either as taking the coefficient, taking the, the comological, the part of comological degree d at the very end, or we can also divide up the d's, d equals summation dv. There's also some technicality with regard to the parity condition that relates sigma, d, etc., this mod 2 condition I had there. That, that also has to be applied for each vertex. So you really have number vertex, number vertices, different parity conditions. Again, you shouldn't interpret this as a natural formula. I mean, what, what I've said is almost complete description, but no, it is required for the individual vertices. Yes. Yeah, I mean the modified ones because I need the marked points. Yeah, no, no, no boundary. Like really what the sum will be is I've written it as a graph here, but it will actually be of the form summation, gamma, graph, maybe this, this extra data here, but then each term, the thing I've written here will be iota, gamma, lower star of some polynomial in cap as in size. I don't want to put the boundary in again. The, the, the, in a term with a corresponding to given stable graph, I want basic classes which are in precisely that, that boundary stratum. Yeah. Yeah. You, I mean, I'm multiplying the contribution from all of this. Yeah. I mean the, the actual form of this will look like, this is a very common form for formulas in topological ring will be summation over stable graphs should be like one over automorphism of the graph somewhere. And then here I have a product of various local factors. Local meaning given by the edge data, vertex data, or lag data in, in the graph, what I have here, just product of all those factors. And so what am I putting in here? Iota gamma is from some product m g v, n v bar to m bar g n. So the things I put in there is they are attached to specific vertex, but they involve like the side classes which are also along an edge. So like the, the edge contributions, if I wrote this out carefully, would be part on one vertex and part on another vertex. Okay. Again, I don't want to get too much into, into the details here because we'll see a sort of more universal language for talking about the sort of formula. But this is the idea and it shouldn't actually be that surprising to you that, that the formula should look like this. Because if you, if you believe that these are all the relations on m g n, which again we don't know, if you believe that those are, are all the topological relations on m g n, and if you take a relation on m g n bar, well if you're restricted to the interior, the principle term, that should be one of those relations or a linear combination of them. That's this principle term. You can also do tricks like restrict to a boundary divisor and then restrict to the interior of that boundary divisor. That should, if you start with a relation, give a relation on, in general, some product of m g n and you would expect that you would get these f z type relations again. So because of that, you should expect that not only the principle term, but also all the boundary alterations should be related to f c relations in some way. If you believe those are really all the relations. Yeah, so, so the reason why this is contradicting each other is the details of what happens when you restrict to some boundary stratum. That you aren't just like picking out what's in the center, you're, you're multiplying by some psi classes more or less, because if you restrict to given boundary stratum, then lots of different components from other boundary strata will contribute. You have like a sum of a bunch of contributions which should fit together to make a natural f c relation in the end. And one of those things is multiplying by some psi classes. And that's why you start with things which aren't in the range, which would give relations, but then you multiply by psi classes to add some corrections from other boundary strata and you end up with relation. Okay, so that, that's sort of, maybe this is historical note about, about how, how one might guess this formula. If you're stranded on island and want to remember the f c relations and want to know what the relations on m g n bar is, you, you can work out what they are by thinking of, looking for something of this general form, which actually restricts nicely to these relations on the interior of every boundary stratum. Yeah, this is what I was saying before about these aren't really f c relations, they're given by the same formula, but they aren't in this range. This is only zero if I need size of sigma plus summation a sub i is less than or equal to 3d minus g minus 1. That inequality isn't, isn't required to, to hold when you distribute sigma and the degrees here. Right, so as, as I said, I will give a sort of better language for writing down this formula. It would be quite, quite annoying to write down the formula at the moment with what we know, but I want to talk now about co-mlogical field theories, and give and tell our matrix action on them, which it turns out is really the natural way of thinking about these relations. I'll always just turn this to coft. And I won't talk about these partly because they're the natural way to discuss these relations, but also because they're sort of, co-mlogical field theories are really ubiquitous in families, when you, whenever you've taught a logical classes on MGN bar, it seems that a lot of those come from different co-mlogical field theories. So these definitions and techniques are really important if you want to think about classes on MGN bar. So what is a co-mlogical field theory? It's, I have to state a lot of, a lot of data usually call our co-mlogical field theory something like omega. And at its base, this is some family of homology classes, of co-mology classes. So for every G and N, we should have some class. I'm also going to, there's also additional data, which need the data and, yeah, sorry, greater than zero. Yes, the same range I'm always talking about MGN bar. So what is this? This is really going to be a co-FT with the unit. The unit being the one at the end here. But what are all these things? First, V should be a, I'll say a finite dimensional Q vector space. Ada should be a non-degenerate metric bilinear form on V, so metric on your vector space. And I'll often use the notation Ada inverse to talk about the bi-vector, this is the element of V tensor V. Ada itself is in V. Dual tensor V dual, inverse should be in V tensor V. So if you want to think about coordinates, this is given by the inverse matrix of the matrix defining Ada. That will show up later. Element one will just be a distinguished element of the vector space. This is sort of the numerical data, this vector space in bilinear form. Forms the base of the co-mlogical field theory. And then these elements omega GN should be an element in the homology of N bar, rational coefficients. You could also define child field theories if you replace homology by child, but much less is known about those. So I'll put co-mlogical field theories here. I want to tensor now with take the dual of the vector space V and take N copies of that one for each marked point. So we usually think of this as it's a multi-linear function which accepts N vectors in your vector space and outputs in co-mology class on MGN bar. Again, for any GNN, you have such an omega GN. Okay, so to be a co-mlogical field theory, you just had us by, I guess we'll group them as three axioms. More or less the way you should think about this is that it's a family of co-mology classes on modular spaces of stable curves. For every MGN, you have some co-mology classes which are coherent with respect to the maps between the MGN bar. So first axiom is that MGN is invariant under the action of spectrogroup Sn acting simultaneously on both factors. It can relabel the N marked points in your giving an automorphism of the co-mology of MGN bar, and it can also permute the N factors here in your tensor product of N copies of V dual. It would be invariant under that if you're thinking of it as a multi-linear function on product of N copies of V two there then should be Sn act covariant as a function there. Second, we want it to behave nicely under pullback by gluing maps. So to say there are two types of gluing maps, maybe I'll just state this for case where it's simpler to say. So for if you have the gluing map from MGN plus two bar to MG plus one N bar given by gluing together two marked points, say the last two, we want to take the pullback of some class on MG plus one N bar plus we're going to take, I'll take omega g plus one N evaluated at some simple tensor let's say one tensor V two. So this is some class in the co-mology here and pulling it back to class there and the result should be equal to omega g N plus two. Now I have I can give it the same inputs I gave before but I need to give it two additional inputs. Additional inputs I give are precisely this by vector given by the inverse of the matric. So if you prefer you can think about this as elements here as you're taking on the right hand side you want to contract your tensor product of N plus two copies of V star by this by vector to end up with only N copies of V star on the other side. And I'll just write and similarly for the other gluing map. For the other gluing map you have two different curves gluing together one point from each so on the right hand side you'll have the tensor product of two copies of two things which look like this in your eta inverse by vectors or distributed over both pairs. It's the natural thing to do given this definition. The part two is saying that the co-mological field theory should behave well under pullback by gluing maps and behaving well is determined by this by vector contraction. The final condition you might be able to guess is about behaving well with respect to pullbacks by forgetful maps. I from mg N plus one bar mg N bar is forgetful map we want pi star of omega g N v one through v N something now we need to come up with one additional input to give omega g N plus one sorry into commas I should probably be consistent use tensors and well fortunately we have this this unit element. There's one last thing which it should possibly be separated out from three but it's really spirit doing the same thing which is that number m zero two is not it's not a thing we want to talk about so we can't pull back from m zero two to get m zero three so we can think of this as a formula telling us how to evaluate an insertion of the unit the unit acts as pullback but what if we wanted over here to write omega zero three of v one tensor v two tensor the unit so this isn't going to be pullback of omega zero two because we don't have omega zero two instead this should just be the metric applied to pair together v one v two so all right so these are the three axioms of a commutical field there right yeah I mean in in in theory you could you could wonder is there a formula like this for omega one one of the unit um the way to think about that would be you can compute omega one one of the unit by using the other formula since you can get m one one by gluing together two points on m zero three I mean m zero three is really special because it's only zero the dimensional modular space that we talk about data should be non-degenerate no so no though we'll mainly be talking about I mean get this in second so this is the basic definition but there are a few basics I'll give a list of some some examples of commutical field there is in a second yeah dude this is this is all you you want um in practice eight of one one will be non-zero in all the cases to I guess it will often be zero no eight of one one can be zero certainly um this will become clearer once I use these definitions to define an algebra structure on v so you know I'll I'll get two examples of commutical field theory in a minute definition quantum multiplication on v is defined by let's just say the beginning it's commutative associative um multiplication operator by star on I already said it was on v three determined by so if I take data of v one star v two with v three so if I tell you what this is for every v one v two and v three this will uniquely determine the multiplication because it is non-degenerate this is just going to be omega zero three of v one tensor v two tensor v three interpreting that as a rational number because omega zero three is just because m zero three is just a point it's a homology class on a point d sure I mean I was saying a as a because the fact that's commutative in the fact that's commutative is obvious from the fact that um I mean it the fact that's commutative is obvious from definition the fact that because we need to use the s three symmetry over here v two and v one without changing the value associativity is is a nice little exercise to do because you're actually going to need to use parts two two and three here more or less associativity here corresponds to looking at m zero four and using the um relation between any two points and m zero four are the same some of those properties you can prove associativity that way it's it's definitely not obvious from this definition it's implied by what we had before I should say that by three rather the last part of three one this unit element is in fact um they under this multiplication say this differently as we we choose one to be our end element this is a turns v into a commutative out associative algebra with the unit and it's actually more than that v becomes a Frobenius algebra because you have some extra structure given by how you define the multiplication here these the quantum algebra or Frobenius algebra of the homological field there right I mean Frobenius algebra more so from this definition you can see that um eta of v one tends v one product v two with v three equals eta of v one product v three with v two Frobenius algebra is more or less just a commutative associative algebra with a metric on it satisfying this this identity with respect to multiplication it's a Frobenius algebra of v two of t omega all right so you have this algebra structure which justifies why i'm calling this thing one they have some examples first example is going to be quite simple v is going to just be one dimensional eta is going to be one um of course one to matrix one the unit will actually be one and i just have to tell you what the classes are so these one dimension omega g n should just be an element in the comology of m g n bar and define it to be the total turn class of the hodge bundle hodge bundle i mentioned a couple days ago it's just the um vector bundle of one forms on your curve sections of the dualizing shape in general and so checking this is a comological field theory is just a matter of checking that the total turn class of the hodge bundle behaves nicely under these pullbacks basic properties of it i know that you have to use the the total turn class if you use like just the top turn class that that won't work because this gluing map here that have pulled back by changes the um this is an example of how um most comological field there is are going to be um not of pure degree there are mixed mixed comological degree usually certainly don't have to be but this is an example of why they have to be of mixed degree you can't have something always be of degree g because then when you pull it back from genus g plus one genus g the comological degree doesn't change but the genus does change this is this is an example examples hopefully convince you that there are at least some simple examples of this you can also modify this in a few ways to write down other other like one-dimensional comological field theories in terms of the turn classes of the hodge bundle now some more complicated examples i mean if you're in more than one dimension the examples tend to be harder there's the right all the details in this one makes they don't want to get caught up in defining the verlanda bundle but if you take the total turn character of the verlanda bundle or if you prefer to call it the bundle of conformal blocks then this will be a comological field theory and i should say so in the first case of course you can guess what the quantum algebra is the furbenius algebra it's just q nothing else that can be in this case though i mean i haven't told you to define verlanda bundle you need various input data like choosing a choosing a le algebra choosing a level the algebra though is going to be what's known as the verlanda algebra the fusion algebra that's a much more complicated example yeah so usually people were working with bundles conformal blocks for linda bundle if you just look at the bundle itself and think about pullbacks and satisfies various like factorization roles the bundle itself that implies the same that implies the comological field theory properties for the turn character of it turn character is basically a multiplicative function of the bundle third example i want to mention is actually technically not a comological field theory as i've defined it but it's almost one so from affliction theory of the target space x choose any x you want let's say so this gives say what i mean by gives it gives a comological field theory put quotes around it because it's like a comological field theory with coefficients so this notion of comological field theory it's coefficients it won't show up but again what i'm talking about but it's a variant on this so you can do coefficients in the novical frame power series in beta or really e to the beta where beta are effective curve classes on x so what's going on here is by coefficients you basically have to tensor all the definitions by some q algebra in this case the novical frame and so you can think of it as your function omega g n where you insert n factors it will output not a rational number but an element of this novical frame and yeah novical ring i i again don't want to get too caught up in defining this but it's and maybe i should say e to the beta beta corresponds to effective curve classes so this is basically taking the free monoid on the the i won't take the i mean it's a vector space with basis corresponding to the effective as a vector space it corresponds to the basis corresponds to the effective curve classes on your target space and the way you define this is for each curve class you have some space of stable maps to x with with image in that curve class and if you take the virtual classes for that for all curve classes beta and some of them together in a generating function maybe capped by some comology classes and then pushed forward to mg n bar you end up getting a homological field there in the sense with coefficients i should say here that b the forbenius algebra ends up being the quantum comology ring so the inputs that you give are comology classes and then you have some product role with coefficients in this novical frame all right so those are some examples to convince you that there's some geometrically interesting homological field there is i will give at least one more example tomorrow i also want to say one more construction you can think of another way of constructing comological field theory so given a homological field theory omega they may on the pair v eight of one that basically numerical data lowercase omega g n of some input factors that be the degree zero part of the capital omega g n so again in general this is some impure um comology class might have contributions in every comological degree just take the degree zero part then you can check because degree zero things pull back to degree zero um then little omega is also a comological field theory with the same other data v eight of one and has with the same with the same algebra structure the same forbenius algebra so this is now a comological field theory it's only in degree zero so it's not not really about comology so these are often called topological field there is a such degree zero f days are called topological field there is any homological field there is an underlying topological field there all right so in a couple minutes remaining i just want to say two more things first is important definition which is so we have this algebra here you can ask when is this algebra semi simple or maybe when is it semi simple after some base change and in that case you if it is semi simple you call the entire comological field theory semi simple write that eco of t is semi simple venous algebra is semi simple and then reason why we care about semi simplicity is tell them on more or less this is a statement of some of what he proved and proved more than this um this is also known as given tolls reconstruction conjecture and it's saying that it right now is given tolls or matrix action acts freely and and transitively on the ofts with a given semi simple for being this algebra i haven't defined given tolls or matrix action to an action of sometimes known as i guess the symplectic loop group um some explicit matrix action which once you write out definition will look very similar to the sort of graph some of those writing earlier about the relations on mg n bar how to get those from the fc relations our matrix action is doing some operation similar to that on comology of mg n bar it's an action comological field there is and the term of tell mom says that if you've two comological field there is with the same propenious algebra equivalently the same degree zero part that's the same then there's a unique our matrix taking one to the other and the result this is essentially is a classification of semi symbol comological field there is i'll write that what is this classification and even be very explicit any comological any semi symbol comological field there it can be written uniquely in the form are applied to a another into any comological any semi symbol comological field there a capital omega is given by applying some matrix action to its underlying topological field there it's equivalent here yeah yeah so if you think about how i defined the quantum multiplication i defined it just using omega zero three which is only in in degree zero so that gives one the that gives one direction the equivalency other direction is that you can recover any any degree zero information from the m zero three and that's just a matter of taking basically degenerating any mg n bar into lots of copies of m zero three bar okay that that will do it for today the plan tomorrow the last day is the outline how to use this this theorem of telemon classifying semi symbol comological field there is to actually get topological relations and explain how the relations i outlined before coming from the fc relations sort of naturally pop out of this r matrix action if you choose the correct comological field theory and look at it in the correct way to get topological relations but i should say this is a general tool the theorem that can be used to get explicit formulas for any semi symbol comological field theory at least in theory there's a question of how do you know what the r matrix is and there's a sort of second part of telemon serums and there's certain conditions the r matrix is determined by some differential equations actually in a lot of cases effective way of getting formulas for comological field theories again provide their semi symbol