Geometric Recursion with a View Towards Resurgence
Andersen, Jørgen E.
Borot, Gaeton
Orantin, Nicolas
We shall review the geometric recursion and its relation to topological recursion. In particular, we shall consider the target theory of continuous functions on Teichmüller spaces and we shall exhibit a number of classes of mapping class group invariant functions, which satisfies the geometric recursion. Many of these classes of functions are integrable over moduli spaces and we prove that there averages over moduli spaces satisfies topological recursion. The talk will end with a discussion of possible resurgence perspectives. The construction of geometric recursion and the results relating it to topological recursion is joint work with Borot and Orantin.
 This is joint work, most of it, from the start on. I will say when I get to something that is not joint work with Gertrand Boreau and Nicolas Arrentain. So they are both students of Bertrand and the geometric recursion clearly takes its sort of origin in topological recursion, but it's quite different from topological recursion and you will see how. So let me just start by looking at a setup that I would like to consider. So this is a very general setup, so I want to look at a category S, which is compact oriented surfaces and the morphisms are just isotopic classes of diffeomorphisms. And the target category would be just be some category of vector spaces. You should think of infinite dimensional vector spaces, most of the applications I have in mind. And then I have a functor given from S to V, so it assigns a vector space to every surface in a functorial way. And moreover, I want a functorial assignment of vectors inside these vector spaces for every surface. And then of course, since it's a functorial assignment, what will happen is that these vectors here will be in the mapping class group invariant part of these vector spaces. And so you might think this is a very particular situation, but actually there are really a lot of examples that are very naturally of this form in low dimensional geometry and topology. So let me, that's what I say right here, and let me start to give you the first few examples. So the first example, I will simply take E of sigma to be continuous functions on Teichmann molecular space of the surface. Okay, that's the functor E. And then of course, I think you will certainly agree with me that if I take the function one in every Teichmann molecular space, then that works. Now you may think this is an utterly trivial example, but I will get back to this and it relates very closely to Mia Sakane's work of Mia Sakane-Makshane's identities and her work on volumes. Okay, second example more interesting. So let's take S to be the set of multi-curves. So this is the set of isotopic classes embedded close one manifold in sigma such that no component is isotopic to a boundary component nor are any two of the components isotopic. So this is first on multi-curves. And now I define a function on Teichmann molecular space also. And what it does is that it sums through the set of multi-curves. You tend to take the product over the set of components and then you apply some function which is sufficiently fast decaying function at infinity to the length of each of the components and then you take product or components sum over all such multi-curves. So obviously, this is if the decay is fast enough, this will be an absolute convergent series and it's a nice continuous function on Teichmann space and it is mapping class could be invariant exactly because if the decay is fast enough, you can reorder the elements here and the set here and that's all that a mapping class would be doing. So I will be specific about this in a second. We will actually look at boundaries and have geodesic boundaries. But right now I'm not so specific. I will come back to this example in great detail. Okay, so basically this list here is just to show you that there are lots of things that actually fit into this general scheme that I started considering. Yes, yes, I will get to all the details of this in a second. But for now, I just want to motivate with some overall examples. But you will see typically we need some kind of topologies in these vector space to make sense of these sums. But okay, let's consider continuous functions again. And now what I take is I take the trace that we saw this morning of F applied to minus the directly the plus built on a differential operator on the Riemann surface where we point in type of space. So this gives again a continuous function provided F is decaying sufficiently fast. Same thing with something more simple. Suppose you take the Vapidas and Simplactic 2 form. Well, that's a 2 form on type of space. If you take Bayer's complex structure that fits in the framework too because the vector space is then just smooth sections of the endomorphism bundle of the tangent bundle. If you take closed forms on type of space, of course, this fits in the scheme as well. You could imagine representing homology classes. We know that we have all of these stable even dimensional classes, but we also have all the unstable odd dimensional classes. And we don't really know how to construct form representatives of these, right? So that's sort of just a remark for now. Then of course I can consider a different modular space. I can consider the modular space of G flat connections on the surface where G is a some semi symbol E group. We could rather be complex or real. And I could then take the Fogg-Rossley Poisson structure on the space. That would be a section of the second wedge product of the tangent bundle of this modular space. You know what? Rossley is visiting our center. And one of the things he needs to do is present gain with a copy of his passport. And this is copy from his passport. So all right. So the next example is the narrow seamen's chichatry complex structure on the modular space of these flat connections. Now of course in order to get a symplectic situation, I have to fix conjugacy classes at all the boundary components. And I require that the holonomia contained in those conjugacy classes. And then I really get a nice, so this one is I think dying, but you know I get in this case here a nice vector space which is now more complicated because it is smooth functions from type of space to the space in which complex structures live in, namely smooth section of the endomorphism bundle over type of space. But the whole thing is equivariant, right? So it's really a mapping class group invariant element in this thing, this narrow seamen's chichatry family of complex structures. Okay, so along the same V, I can do. No, and it stops at this slide here, but let me just mention them. So there is the Ricci potential in modular space. There's the hypercalar structure of Nigel Hitchin and the Higgs bundle modular spaces. You could even talk about representations of mapping class groups. Itself, they will just be flat connections in trivial bundles of a type of space. You could also talk about boundary vectors in TQFT. They don't quite fit the scheme here because in this case here the boundary vectors are only invariant under the diffeomorphism with the extents over the three manifold. But okay, let's mention them anyway. In fact, invariance of both three and four manifolds, smooth four manifolds, can be fitted in this scheme simply by representing three and four manifolds by either HECA diagrams or trisection diagrams. Also forms that come out of Grumman-Witton theory and amplitudes in closed string theory as far as you believe that they are top forms on type of space that are mapping class group invariant. So tons of it. Okay, let's now be more specific then. So the category I would like to work with is compact oriented surfaces of negative Euler-Catzuristic with a marked point on each boundary component together with an orientation of the boundary in such a way that it splits to in and out. And the inclusion of the ins in the surface gives you an isomorphism on pi zeroes. It means that there is exactly one in for each connected component and all the rest are out. Okay? And then isotopic classes orientation preserving. The amorphisms which preserves the marked points and the orientations on the boundary, modular isotopy which preserves all the structure. Those are the morphisms. So this is the precise category of surfaces we want. Okay. So now let's go to the category of vector spaces that we want. Well at first go what we would like to have is house of complete locally convex topological vector spaces over the complex numbers. So these are vector spaces whose topology is given by a semi-norms. And then I want for morphisms, I just want morphisms of locally convex topological vector spaces. But we have to go a little bit more complicated than that unfortunately. And let me just sort of preempt the example of Teichmann space. So example of continuous functions on Teichmann spaces. And so the point is that, you know, I'm also going to introduce precisely which Teichmann space I want in a second. But for now just think of Teichmann space with metrics that are hyperbolic on surface with boundaries such as the geodesics, so the boundary of geodesics. Then of course inside here, you see, and then I might just, so the vector space I just want to consider is the one that we saw before that's just continuous functions on this Teichmann space. But in order to somehow get the whole analysis to run under the recursion, I have to consider, you know, pick an epsilon and then consider the subspace of Teichmann space that consists of surfaces with all closed symbol geodesics of lengths at least epsilon. So I need those also. And so therefore that epsilon will lie in the real line plus, and I will have C0 of these guys. So what happens is now that I don't have just one vector space. I have a system of vector spaces indexed by a directed set. So that's why I now switch to saying that an object in my target category is a directed set and an inverse system of such house of complete locally convex topological vector spaces over this directed set. So here this means, you know, smallest geodesic bigger than or equal to epsilon. So initial space is directed limit of that's right. And therefore you get such in the directed system, so the continuous functions. And now I also want some specific set that indexes the semi norms. And in this case over here, you should just think of this epsilon here as well a set of compact subsets of this system, I mean this set of system at least epsilon, right? So then if I take the semi norms, which is max of the point wise norm over those sets, I will get the appropriate semi norms to get the natural such vector space structure on these two. Okay, so that we have here, so we have these projective limits like this and then of course what I can do, yeah I can take the projective limit of the whole guy, that's this space here in this case here. And then of course what I can look at is I can look at sort of the subset that has finite norms where the norms that I'm, so semi norm where the norm is simply the supremum over all compact subsets. So this subspace prime here just means functions that are bounded on all the system sets. These are epsilon system sets, right? So the semi norms here, this guy, unfortunately this doesn't really work for that guy. You know that's the max norm over that subset. So not all the continuous function of course has finite norm for that, obviously, it's only the bounded ones. Okay, and then morphisms is completely naturally defined, the morphism of such two objects, you know is an inverse system of continuous linear maps, phi ij, where i is running through the first index set and j runs in the second one but it is less than h of i, h of i is some order preserving map from one to the other and this should of course induce continuous maps of the limits in such a way that these prime spaces are mapped to each other. Okay, very good. So I assume that I'm given, you know, a fountain from this category of surfaces that I introduced to this category C here. So such a C I call, you know, a target theory. It will have to satisfy a few more things in a second. But anyway, the whole idea is try to recursively define a vector inside the e prime guy of the surface, mobbing class group invariant part and we want to do this recursing in the Euler characteristic. The basic idea is to remove a pair of pants from the surface and this will increase the Euler characteristic by one. I'm only considering negative Euler characteristic surfaces so it will end with Euler characteristic minus one which is either a pair of pants or one whole torus. And so over on the side here try to indicate the three different types of pair of pants and binnings you can imagine. Namely one is where there are two boundaries in common with the surface of the pair of pants and the other two, so that's the B case and the other two is the C case and the C case we only have one boundary in common and the two other boundaries are inside the surface. Okay, now this requires me to have further data given because this means that, you know, if I want to define something for disjoint union I would like to have a bilinear map and morphism here from the cross product two vector spaces for the two individual components over to the disjoint union and also if I have gluing maps, you know, if I have two surfaces and if I have a pair of boundaries that I want to glue on I need to have gluing maps because then I can somehow, you know, cut out this piece here each time and I can stick in some starting data say A for a pair of pants, Mapping Class Group Invariant part of that and D for some guy for the one whole torus also Mapping Class Group Invariant part of that and so I will start by defining these two to be that initial data and then I need recursion kernels so these are just elements also in the vector space for a pair of pants both B and C and B, well B depends on a choice of one of the boundaries on the pair of pants that namely goes to the other one, right? So, post here this is the minus boundary and the pair of pants has to have its minus boundary go here but then the pair of pants has two out boundaries and I can decide whether it's one or the other that goes to this boundary here. So therefore I have two copies of B. Okay. All right and then the idea is simply I just sum over all possible ways of extracting pair of pants of the two kinds of types, I stick in the recursion kernel in the first slot and I use the morphism for gluing and I recursively assume that I have defined the thing for what remains after I extract a pair of pants and I sum like this. Now of course this has problems because these two sets here of isotopy classes of embedded pair of pants these are of course infinite, I mean they're countable but they are infinite sets. So therefore I will have to do some kind of analysis in order to understand what this sum means. Now if it could actually be quite interesting to maybe just do some kind of divergent series here where you just assume that these terms here are bounded by something for each extraction and then you would get a divergent series and you'd have to work on this sort of the way that we are having the main theme of this conference. But what we will do at first is actually somehow specify conditions on these gluing maps and the initial data in such a way that these sums here will be absolutely convergent. Okay. So let me now describe what we need in order to make sense of these if we want them to really be convergent series. And so in order to make them. It's not that you are summing over an infinite set. Yes I am. Unordered infinite set. Yes, that's right and therefore we better make sure that the series are absolutely convergent so it doesn't matter which ordering we choose. Right? Yeah. I agree. Okay, so now what we are going to do is actually give the precise definition of what I mean by a C-valued target theory. So this will be a function E from S to the category C we have previously defined. These directed sets of vector spaces with all these semi-norms on. So that's what I'm saying here. So I have the vector space for every surface and I have all the set of semi-norms indexed by in this case here it was all the compact subsets of the system sets in this example. Right? Right now I'm just thinking abstractly there are just some abstract given set of semi-norms. And now I take again inspiration from this example here. So if I have a compact subset of type-molar space and I have a given curve on the surface what I of course I can do is I can talk about the length of this curve because I can just take the minimum of its geodesic length over this compact subset. So we see that there is a natural way that what parameterizes the semi-norms in this example also parameterizes length functions. And so therefore I'm simply just going to say abstractly well I just want abstractly length functions which are defined on the set of multi-curves on the surface. They take value in non-zero complex numbers and they are exactly indexed by the indexing set for the semi-norms. So that's just a natural abstraction out of what I see in the type-molar case. Okay. Now I need that these length functions satisfy three different actions for this theory to work and the first one is polynomial growth action. And so what you do is you say let's take an i and let's take some semi-norm and what I do is I also take a large l and then I look at all the multi-curves that has length less than l for that set of semi-norms. For that semi-norm sorry. So this is a finite so what we need to assume is that this is a finite set and its size should not grow more than polynomial in the length l. So that means that should exist mi and di such that this guy here when I take supremum over all the things that index the semi-norms for the i subspace should be bounded by this. And so in the type-molar case this is simply just a result of ravine that says that honor system set the number of geodesics of length less than epsilon sorry length and l sorry and it's simple geodesics. I'm talking about simple closed geodesics grow at most like a polynomial. I think this di here is 16 minus 6 plus 2n or something like this is the best constant you can choose in that case. But we just assume abstractly that there is this property here for our length. Okay then there is a lower bound and that is directly from the system definition here we strapped this out so it says that for any i there should accept an epsilon i bigger than zero such that the infimum of the length over the set here of all the semi-norms and all the simple closed curves on the surface should be bigger than epsilon i sorry epsilon i. So that's just the analog of saying that I have fixed system sets. Okay, then there is the small pair of pants action and it simply says that for each i there should be some constant such that if I look at all the semi-norms guys and I look at the length and I look at all pair of pants embedded in the surface for which the length of the interior boundary of the embedding is smaller than the length of the boundary that's common with sigma then this should be a finite set and bounded by Q i. So that this is satisfied for type mirror spaces follows by some work of Hugo Parlier. So those are the three things I would like to have polynomial growth action I would like a lower bound action and a small pair of pants action. So now I specify the gluing data so again I want this way to take disjoint unions so a linear map here that satisfies nice associativity when you do Cartesian products and associativity of unions on this side. I would also like to have gluing maps like this when I have pairs from the two boundaries from the out of this and in here and then I make the gluing and so I'd like to have linear maps from the vector space of these two to the glued thing and of course this should be compatible with gluing morphisms and with associativity of gluing and with disjoint unions. So this is exactly what I said before here that I was in need of in order to make sense of this. So very good. So now what is the initial data that I have to start with? Well as we discussed before I certainly need the thing for a pair of pants but I also need the recursion kernel, the C1 and BB I need also a pair of pants where B is one of the out boundaries of P and of course if I have a diffeomorphism that interchanges the two out boundaries then I must have this condition here on B. And then D and I want to assume that D is a separate guy. I could assume always that C was some kind of trace of D because for one whole torus I could just sum over all possible ways of breaking it into a pair of pants but that would put some trace class conditions on C that we don't want necessary to have in all our examples. So I just allow D to be a separate part of the initial data. Okay and here is the precise analytic properties, the decay action that we want for the initial data combined with the gluing maps. So it says that if you take two guys from the indexing set of the pair of pants and the complement of the pair of pants for some embedding of a pair of pants inside sigma and now you take a K that is less than this map that you get from the directed sets by the gluing morphism and then you take any semi norm in the glued surface then you require that there must exist an SK that is bigger than this DK that went into the growth rate and functorial constants that depends on these indices i, j and k such that if you take and look at the gluing morphism applied to B and any vector in the complement of the pair of pants so that's called, where is it? It's called sigma complement. I get there. So any vector V that lies in the E prime space, mapping class, gluing variant part, you must have that this norm here is less than this constant times the norm of V times this one plus and then you take the length of the boundary in the interior and you subtract the length from the stuff that's common with the boundary and then you add one and you take the plus part of that and the plus part just means if it's negative it's zero but if it's positive it's just equal to the identity. So you only look at the things that give you positive quantities for this quantity here so only the pair of pants for which the length of the boundary is smaller than the interior length and that's typically what's going to happen that's most of them in fact all of them except for finitely many by our small pair of pants action and then you lift, you require that this lifted to minus SK, bounce this quantity here and the same thing for C. So this is the precise analytic condition that we require on this initial data and I'll give you several examples in a second that satisfies this. Okay, so now we start the recursion. So for an empty surface we may want to define this to be just one for a pair of pants and one whole torus we are bound to define this as A and D and now we just use this formula here to define it for any surface of Euler-Cathoristic less than or equal to minus 2 and for connect, this is for connected surfaces and for disconnected surfaces we just multiply them together via this you know disjoint union morphism. And so the main theorem of this is that this assignment is well defined and so more precisely actually this series here is absolutely convergent for any of the semi norm and its limit lies in the E prime guy and it is functorial. In particular this state that we get out of it here lies in the E prime guy and it is Moppen-Klasgrub invariant. Okay, so let me just say a little bit about how the proof goes. So you just for example take this part of the sum here and because we want to show absolute convergence I just take the norm of everything inside the sum and now if I just look at, so obviously if I just take my estimates that I have for the initial data, so I just go back here and show you, just use this estimate here straight off. If I use this estimate what I get is that this side here is bounded by this quantity here where this theta function here is simply just the sum of all the pair of pants and then you take this quantity here. And now the axioms are exactly set up such that polynomial growth axiom plus small pair of pants axiom will guarantee that there exists an SK bigger than DK such that this is finite. And then the lower bound action plus the small pair of pants axiom will actually say that there exists a constant such that this gadget here when you are taking supremum over all the semi-norms is universally bounded. And therefore what you are going to get is this quantity here so that shows that this series is absolutely convergent and lies in this set here and then you have to work a little harder to say that the whole thing lies in E prime but I don't want to go through all the details of the proof. So in other words you can really make sense of the sums here this way here. Let's go through the example of Teigmuller theory and let's me introduce precisely which Teigmuller space I want. So I take, you know, remember that the objects I have in this category of surfaces they are surfaces where each boundary component has a mark point. So I denote those OB here. And now the Teigmuller space so this guy is simply just the usual way of defining Teigmuller space you define it to be diffeomorphism from the surface to a Riemann surface where this is now a bordered Riemann surface, modulo and equivalence relation. And such two guys here are equivalent if there exists a bihologomorphism from the first Riemann surface to the second such that when you do this composition here around this first of all restricts to the identity and the mark points and it is isotopic to the identity on the surface via diffeomorphism which restricts the identity on these mark points so it doesn't change these mark points. So it isn't quite the usual Teigmuller space. The usual Teigmuller space you wouldn't have the mark points right? But the thing is that of course you can just forget that you have the mark points so there is a projection from this Teigmuller space here to the usual Teigmuller space of bordered surfaces. And this guy here is simply an R to the number of boundary components bundled over this because when I do the twist around the boundary you know I can keep doing it and even if I do it too pie around I don't come back to the same metric because it's twisted right? Okay now of course I have the group delta sigma which is all the boundary parallel dean twists they act freely on this guy here and if I divide out that subgroup here of the Mopin class group acting on this Teigmuller space that should have been a P here. It's this guy here. Then of course this just drops down to be a circle bundle over I mean a torus bundle over this guy where the torus is this u1 to the number of boundary components. And I need exactly this guy here because I want to integrate over these fibers here in a second when I do the gluing morphism. Okay so you know if we look at a pair of pants what do we have? Well the usual Teigmuller space just says that it's R plus to the 3. Just think of representing by hyperbolic metrics. The boundary components are geodesics so therefore you just have the three lengths of the boundary components for pair of pants that gives you the usual Teigmuller space. Specifies points uniquely there. And the point is that with these guys here you just add a copy of R for each of the boundary components and if you divide by these dean twists that are parallel to the boundary you just have R plus cross u1 to the 3. And then we can of course imagine having coordinates so these are the lengths and these are the twist parameters on this Teigmuller space here. Okay so in a second I'll give you initial data and this means I just give you functions that are defined here on these six variables. In fact most of the time I'll just give you one that is defined here and then I just put it back via this projection that forgot about the mark points. Okay so now let's discuss how are we going to do the gluing maps. And so let's take two guys and let's take some pairs of boundary components we're going to glue on. Then of course what I can look at is I can look at the subset of the Teigmuller space of the disjoint union where I just look at the subset where the lengths of the boundaries that I'm going to glue on agree. So I'm ready to glue right? So that's the subset of the Teigmuller space of disjoint union and now I can just glue so I get a map from here to the glue to the glued surface just by gluing the hyperbolic matrix I can do that if the lengths on the boundaries agree that I glue on. And the point is that this will give, realize this space here where I divide by you know if you take the, if you take the, if you have such a curve that you're gluing on if you do a dint twist on one side and the inverse dint switch on the other side before gluing this really matters but after gluing it doesn't because the two are of course isotopic to the identity after you glued so therefore you can divide that group action out it won't change the image over here and the point is that what you're left with is again just a circle, a torus bundle over this guy here. A copy of the circle for every boundary component, a pair of boundary components you glue on. Okay so now let's introduce these subsets. So sorry what I called here T epsilon sigma is actually called K sigma epsilon here. So it is just the subset for which the system so the length of the shortest deodesic on the surface is bigger than epsilon I take indeed just continuous functions on that. And then of course as we just discussed over here we have a family of semi norms by just a thinking of compact subset here that gives me this locally convex house of complete topological vector space structure on these and of course if epsilon is smaller than epsilon prime I get restriction maps that go this way here. So I have a system I have a directed system as I said I would like to have and now the vector space the total vector space we're considering as of course this continuous function all a type molar space we look at this these norms here they are just supremum norms over the sys-tool sets and so these guys here are just functions that are bounded on the sys-tool sets. Okay yeah then we define a length function and this length function is just take the minimum of the length of a given curve over the compact subset. Now this is actually commensurable to if you just took one point of this compact subset because there actually exist a constant between zero and one such that you know this length here bounds Ck times the length of the all of the elements in the compact subset. Okay now so because this guy here is exactly defined in such a way that it is sys-tool sets of course it satisfies the lower bound action and then as I sort of promised in the beginning here or mentioned in the beginning right there is a result of ravine which actually was refined by Miesa Kani but I think the way we just use it here is just the stuff that was done by ravine that says that you know if you have a simple closed curve on the surface or a multi-curve then if you look at the ones that have growth that have lengths less than L then this here will grow slower than the power of L when L goes to infinity. So the polynomial growth action is also okay and there is this work of Hugo Parlier we could use to show that the small pair of pants action is also okay. Very good so now I have to give you precisely what is the gluing morphism. Well for disjoint units are totally trivial because the tight molar space of a disjoint unit is just a cross product so you have a projection QI to each of the factors and you just pull back with respect to the QI's and you take the product that's the disjoint morphism, disjoint union morphism. Now the gluing morphism is similarly simple what you do is first you take the disjoint union so you get tight molar space of the disjoint union then you look at this inclusion map which is just the things where you say the ones that I want to glue on has to have same length that's a sub-variant so you just look at the inclusion of that and you pull back the functions to there and now you are on the circle vibration or sorry I keep saying circle vibration I mean torus vibration because you might glue on more boundary components than one and so you have this torus vibration and so you simply just integrate with respect to you know the rotation invariant measure on each of the fibers of this map so that's the gluing morphisms everything is very explicit and simple. So initial data means that I have to give you two functions A and C of the six variables which are symmetric in the last two variables I have to give you two B's depending on which of the two output boundaries of the pair of hands I select to be special and so they are just completely you know general functions there are no symmetry of those however they are related to each other namely they if I permute the last two coordinates I go from one to the other. Then I must specify a continuous function that is mapping class gluing invariant on tachymorla space of this of the of the one hole torus and now the admissibility of the initial data simply becomes very simple in this case here it just says that for all S and epsilon bigger than zero there must be exist a constant that depends on these two such that if you take the supremum over the sysdel set of one plus this positive part of the interior length minus the exterior length and you take that times the function for the recursion kernel B this must be bounded by this constant and correspondingly for C. Okay. That is ABCB zero six A constraint. No they are completely free. That's right. Yeah but that's because I have a single out in boundary and all the rest out. If I want to have it the result completely symmetric in all I must satisfy relations. Because the free theory is easier to talk about I somehow I could talk about that and we know what the relations are in this language here. Okay but I rather give you some examples. So let's consider the following. So we call this the Mieser-Khanem-Makshane initial data and so what you do is you take A to be the constant function one and then B and C are given by these very explicit expressions you see here and then D is just obtained by taking what I would call the partial trace you just some C over you know here you insert the length of the boundary of the torus and here you insert the length of the of a simple closed curve in interior that cuts this thing into a pair of hands and then you sum of all sorts. And so when you do this then the theorem says that for any surface the geometric recursion applied to this initial data here with this target space continuous functions will give you the function one. So you know you should think of this as a kind of partition of unity okay and you know I would say the proof of this is completely straightforward except for the one fact that namely Mieser-Khanem's generalization of the Mieser-Khanem-Makshane identity and of course this is why we pick these two functions they are taken directly from formally from Mieser-Khanem's generalization of the Mieser-Khanem identity. So the proof is simply just you iteratively each time you apply the recursion you prove that this is the case via her identity. Okay but the nice thing about this is that this has ramifications well sorry you will see the ramifications second I just want to introduce another set of initial data and this is initial data we call the conservative initial data and so A is again the constant function one. And now B is this gadget here so remember the plus is just the positive part right. So you just take these very simple expressions here and for C you take this and for D you take the sum here and so the theorem now says that you can run the geometric recursion for these and then you will get some functions on a Teigmaler space which I'm up in classical invariance of course there are continuous functions on the modularized space and in fact they are integrable with respect to the volume form that there is associated to the vapid and symplectic form and if you perform these integrals here what you get is exactly just these evaluations here of PsyClasses and so this is the reason why we call it the conservative initial data. And so you didn't consider that. So that's the same with this Mieser-Khanem and McShane didn't consider this either. You did consider this I think. So now I want to generalize these data. So what I do is I look at a continuous function defined on the positive reels and it's going to have sufficient fast decay at infinity we'll get to that but my aim is to say I want to make this sum over simple curves I keep saying that but it's really multi-curves right? Sum over multi-curves and then this product of f applied to the length. And so if you define SF to be the sort of fastest I mean the polynomial decay rate of the function so you assume that this guy here is finite or actually sorry you just defined it this way here but if you know that you have a surface for which 6g minus 6 plus 2n which is this growth of length of jesics that are simple and closed is less than SF then this function here will be an absolute convergence series which defines a continuous function on this type of space. Okay and now for so therefore for example if you assume that SF is infinity then this works for all surfaces and one example of that is that if you just take f to be sort of exponential decaying then of course it's fine. So for disconnected surfaces I just simply define the function for the disconnect serve to be the product of the two components and I for pair of pants of course we observe that this function is one because the only entry in this sum here is the empty multi-curve because I don't allow anything to be isotopic to the boundary right? Okay so now the thing is that we can just do this very simple thing of f-twesting initial data we take the miyazakhani initial data but now we do this kind of twisting so for b we simply just add f of l and for c we take c of miyazakhani and then we combine b with f this way here because it has to be symmetric in the last two right so you kind of let to this if you want to combine it with f and b and then you take the product in the end here and so you take af to be 1 and you take df to be 1 plus the result of summing over the symbol of those curves on the torus and so the theorem which is you know less trivial is that when you apply the geometric recursion to this initial data here you get exactly this function here. So this is where geometric recursion is sort of very different from some kind of topological field theory or something like this because if you took this function here on a surface and you caught along one specific curve then this function will satisfy terrible complicated gluing rules because you would want to try to have geodesics match up with directions along the curve you caught along and so on and it's very complicated. I don't know how to do that in fact but this geometric recursion it satisfies very nicely and the idea of the proof is somehow that when you have this large sum you can always fit a pair of pants inside the complement of these curves and since you're summing over everything this is going to work out so there was a very rough schedule that proof but that's the idea that makes it work. Now let's suppose we have some Mappin-Klauss-Kupin variant functions on Taichmuller space and let's further assume that they are integrable when we multiply them onto the very Peterson volume form so I'm going to use this notation bracket 5 for the integral over the moduli space where I fix the length L1 up to Ln which I therefore have as inputs to the function on this side here and then if you apply this to this function here I mean it turns out that if f for example have exponential fast decay then it automatically follows that the function here is integrable with respect to the vape-pedeson volume form and the result will satisfy this very simple recursion here. So you see that this recursion here only involves the initial data that you started with B and C twisted with F of course but then it's just integrations over R plus and R2 plus and of course these formula here should remind you a lot of what Bertrand just talked about topological recursion except that he didn't write down what the recursion is but I did it here and in fact I think my next slide yes indeed is that if you look at this topological recursion and so this topological recursion takes as Bertrand was explaining to us a spectral curve and a one form and a two form and then you can produce out of this these forms that are indexed by G and N and the point about how it's related to you know geometric recursion is the following that if you start with any spectral curve which is ready to be applied you're ready to apply topological recursion to then you look at the set of ramifications points for X and you consider the sort of standard set of coordinates around each of the ramification points now you take V to be the free C vector space on the ramification points then we're saying there exists a family of admissible initial data parameterized by a parameter beta in R plus four geometric recursion with the target theory being continuous functions on Teichmuller space with values in V tensor power the number of components of the boundary of the surface so you can compute the GR amplitudes we call these vectors amplitudes they become integrable over the modular space of the guys where you fix the length of the boundaries with respect to the usual way Peterson volume form and then what you should do is you take these averages of the guy you constructed via geometric recursion here then you do the inverse Laplace transform in all the variables you end up with C eyes set eyes sorry and the set eyes are exactly the coordinates here so you take the amplitudes from topological recursion that is you know computed and you take these iterated residues like this and you get that see beta yeah so yeah yeah okay it's in conform field here I think might be right so yeah so this links up topological recursion with geometric recursion in fact it's saying every time you can do something with topological recursion you can lift this to geometric recursion you can compute amplitudes in geometric recursion and modulo this Laplace transform this is the same thing as output okay now there is a sort of more general thing that you can do namely anytime you have initial data for geometric recursion where these constants that I showed you for the initial data if they are independent of epsilon then the output of geometric recursion will be integrable over the modulized space with fixed lengths with respect to the very Peterson volume form and if you define the WG ends of else to be the average of these guys then this will satisfy this recursion here okay so for example if we go back this is exactly what you saw for these functions here and so by the way this sort of allows you to tell for answer questions like if you ask so I have formulated this for continuous functions but there is an analog of this for measurable functions so for example if you take f to be the function that is constant one in the interval for say from one to two and zero otherwise then this certainly has fast decayed infinity and if you compute this what you will be getting for these brackets here is the volume of the modulized space for which the surface has a geodesic between one and two so answering questions about such statistics of lengths of simple closed curves I think was open and we give this sort of nifty recursion here namely that these satisfy topological recursion okay and so that's a very general thing as this this guy here says this happens for all geometric recursion you can do provided you know that they are insicable over the modulized space okay it is not only about functions so let me give you an example where we study forms so recall that we had this type motor space for a pair of hands was this guy here so I have six coordinates and of course if I take a torus I also have Flenschen-Nielsen coordinates on the on that well I have to choose a curve to do it with respect to right but then I get coordinates like this this is for the boundary and this is the interior stuff and now I take the target theory just all forms on type motor space and I take a to be the exponentiation of this form here so what I would like to call the way the the the way Peterson's symplectic form for this type motor space and B you just multiply me as a Kani initial data onto this guy here and C you just multiply me as a Kani initial data onto this guy here D you take X of the way Peterson's symplectic form and then the theorem says that when you run the recursion for these guys here you get indeed exponential of the way Peterson's symplectic form on all surfaces so that's a that's an example where you're using forms and not just continuous functions okay now we can also work on on measure of each volume so this is some rather recent results to with these gentlemen here and and and you know we've just put a paper in the archive a couple of weeks ago and so now you consider the bundle of quadratic differential so what type motor space of course this guy has a natural norm which is just take Q which Q bar and take a square root of the absolute value of that integrated over sigma that should have been that sigma there but okay and so there are of course local holonomic coordinates on the space of quadratic differentials and so therefore that gives you a notion of lattice points and there is the measure of each measure on the space of quadratic differentials which is simply obtained by counting lattice points and so we normalize it in such the way that the co-volume of the lattice is one and then that allows you to define a measure on the unit sphere bundle of this cotangent bundle of modular space namely if if y is a measurable set inside the unit norm correct differentials here you simply just defined its measure to be well 12g minus 12 plus 4n times and then the measure of each volume of the cone on it of the size here and so this is a mapping class group invariant measure and it's a result of measure and beats that actually the whole bundle here over modulized space has finite volume and so it's an interesting thing to actually try to compute these measure each volumes okay so we do the following take this function here and now take that function and twist the conceivage initial data with this f and then what comes out we will call the measure of each amplitudes their continuous functions on type more space which are course mapping class group invariant and then we can they actually also satisfy that it becomes an integral function over the modular space and so we average this function with respect to the vapedas and form again over this modular space and it becomes a function just of the lengths of the boundaries and our theorem says that this is a polynomial in the allies actually and this polynomial is related to the measure of each volumes via this explicit thing here so it's zero order coefficient is simply just modulo these quantities here the measure of each volume so in other words you know you can construct this function via geometric recursion and then when you integrate it you get this and so since you always have recursion when you integrate so one of the tools that geometric recursion can be used for is to show that something is satisfying topological recursion because we have this theorem that says that if you can find some function that satisfy geometric recursion its average will satisfy topological I come to that in a second so far n is bigger than one here you're equal to but I come to that on the next slide not this one but actually so if you run through the recursion what does the recursion actually say in this case here well if you look at this guy here was a polynomial in fact here is exactly how it is a polynomial and here are the coefficients and so we can write down a recursion that just involves the coefficients so you start out by setting these to be zero you put this to be this delta function and then if one one is this guy here with this C2 of two and then you have this recursion relation here with the B's and the C's and the B's and the C's are given explicitly like this where you see even C's is going in in linear and quadratic way so to answer your question if the surface of genus G and at least some boundaries the measure of each volume is given so this this is what I just said before but it turns out actually that if you have a closed surface of genus at least two then you can actually get the miss of each with no mark points by this coefficient here so you put L equal to one and you look at one puncture and you're correct with this factor so this means that you know we can very quickly you know computed via topological recursion or if you like just using this recursion relation here we can give you all these polynomials they could be put on a computer and spit out as high G and N you want okay so so far all my collaborators have been on board with this and and so this so far I've just given you things we we really have to have results and if you now allow me to speculate a little bit then the next stuff will not so if you if you have complaints about this and if you have some I'd like to hear it but don't blame my co-authors for it please okay so I I just want to try to make some speculations about the closed string field theory so in closed string field theory you have a vertex Hilbert space which comes from conformal field theory right and you have an inner product on it and the whole key about this is that this theory provides you with brackets which is indexed by G and N and some positive real epsilon which are you know multi linear brackets from V to the end to V and all of these brackets should satisfy the quantum master equation now I'm not going to write down what a quantum master equation is but just tell you that I can you know I can turn of course I can use the inner product up here to turn these brackets into brackets like this if I like so now they're just complex valued and I just the equivalent to these brackets here via this inner product like that I just dualized the first variable and then the whole sort of expectation or the thing that you or the definition of whatever you want is that these multi linear pairings are determined by integrating certain top forms over some subsets of modular space MgN not bar here but some subset inside the open part of the modular space and this subsets here together with the natural properties that these forms satisfy when they come from conformal field theory is you know the quantum master equations for these guys here is equivalent to that these subsets of modular space is satisfies this here under of course natural identification of the forms under such gluing so you're supposed to find some subsets somehow such that the boundary of these subsets for Gm is obtained this recursive way here which reminds us a lot about what we've just seen and certainly a lot about topological recursion because these are exactly the terms in topological recursion okay so what I claim is that we can actually generate a function that had that is a characteristic function on the modular space for a set that satisfies this analog of the quantum master equation here and so it's very very simple you just take the following function from r plus to r it's a measurable function it is t in the interval from 0 to epsilon and it is 0 otherwise and I require that epsilon has to be less than ox into one now I take initial data which is just a abcd coupled to these functions here and it is the should have been Mirza County McShane coupled to these f's here okay and now if you for each point in teichmann space define n epsilon sigma p to be the number of simple closed theodistics of lengths shorter than epsilon then theorem and so that one that part is okay that's easy to check actually that if you compute the amplitudes in geometric recursion for this function here coupled to the Mirza County McShane initial data you will exactly get one plus t to this power here and so in other words if I take t to be minus one the result will exactly be the indicator function of some subset and what is this subset well this subset is simply just the systle set just the subset of modular space for which the you know the all the simple closed interior theodistics have length at least epsilon and so if you analyze this function here and look at its derivative in a distributional sense you will exactly see that it's bound these sets here if you take this set here and hit it with require that all the boundaries also have length epsilon then these sets here exactly satisfy the quantum master equation so that of course tells you sorry that tells you that if you want to do this integral here you just take and multiply this guy onto these forms and integrate over the whole space and so therefore I think it's very likely that one should further be able to build the whole form here via geometric recursion because these forms here satisfy what you expect from you know conform field theory they satisfy factorizations and that is sort of how we prove that you know the function times the vapidism-sumplictic form because the vapidism-sumplictic form satisfy natural factorization rules you know we know that that we can do the integrations and we get topological recursion so I really expect that these brackets here will satisfy topological recursion proved via geometric recursion construction of this okay that would mean that we can compute them right so then we should be able to compute these and we have a recursion relation on these and so on so people who know something about string field theories would please tell me about what's what examples needs to be done here because I think this could be doable okay another example which I think one should really look at is a Kalabiyaw many folds so you should take M to be the modular space of complex structures on the Kalabiyaw you take the standard line bundle that this guy has gotten by taking the three zero part of the homology and then you're supposed to have these FGs which are the genus G topological string amplitudes which is supposed to satisfy the BCOV holomorphic anomaly equations like this you should take the mirror and then there should be holomorphic sections of these line bundles which are the genus G Gromov-Witton potentials of the mirror and then the mirror symmetry states somehow that you know FGs can be obtained from some kind of large complex structure limit of the curly FGs and so this has been understood in genus zero but certainly not in higher genus as far as I understand in general yeah yeah yes and so what I would like to understand is can we expect that these FGNs these FGs here can be given by geometric recursion so I'm trying to work on this but for now it's only just an expectation but of course then we can go on and say well Witton really had this very nice intriguing interpretation of the BCOV equations right what he did was he considered this set here which we also saw in Bertrand's talk a lot and the thing is that then this set here satisfies in this case here the heat equation and so of course if you just continue following this analog the heat equation is really a special case of parallel transport equation with respect to the Hitchin connection where you have states in some spaces of holomorphic sections parameterized by some space T which is parameterizing complex structure some pre-quantizable symplectic manifold compact with a line model over it and so I would say that one should expect that this parallel transport problem here can be solved in this form here again where these FGs here should maybe be constructable via geometric recursion maybe one should first go and restrict to the case where this M here is the modular space of flat SUM connections on the surface so I don't know but I would like to discuss this with people if they're interested I can say the following or let me be more precise if you like oops sorry take X to be a compact oriental three manifold boundary sigma consider the modular space of flat SUM connections on sigma take the transimons line bundle take the complement of the zero section of the dual line bundle then you can embed all the sections of L to the K in one and the same space namely functions on that total space that transforms in the case guy for the case power therefore in that space it actually makes sense to sort of consider the total quantum transimons state series like this which is now holomorphic function on the total space of this line bundle here of course all of it parameterized by Teichmuller space and so the conjecture would say that the total state series are resurgent and it has its singularities contained in the set of exponentiated transimons values and the zero section inside the pullback of this line bundle to the SLNC modulized space via the cotangent vibration using non-abelian hodge and then you should look at modulized space of flat SLN connections on X and compute their transimons values and so this is should be where this guy has its singularities so I think tomorrow we will see maybe an analog of this in the case where the surface is empty namely for compact manifolds that Starwars might talk about but I don't know anyway all I can say is that actually this conjecture here really works if you look at the complement of Toros knots in S3 and that's actually related to some work that I have done very recently with my PhD student William Miskard and there we established for cypher fiber manifolds that the Otsuki series is resurgent and that the singularity set is a subset of the classical SL2C transimons values and that part is on the archive this part here so this this guy came out some four months ago William or one yeah so yeah William is here you can talk to him about it Starwars just told me I have to finish very quickly higher teichmuller theory could probably also be used if you want to get at the Selberg trace formula via this I don't think you can cut along pair of hands for the same reasons what you should do is you should develop a theory where you extract triangles instead of pair of hands then you will have corners on the boundaries and this will be kind of an open geometric recursion we're working on developing this and I would think that we can prove after we've done that that the Selberg trace fits in that scheme and we can see that that would satisfy open geometric recursion therefore get same kinds of results on averages and such things thank you very much the problem is constructing a steady structure I would like to hear more details about that in your joint paper that's out already I mean yes okay but you say it you don't think it satisfies yeah questions
 This is joint work, most of it, from the start on.  I will say when I get to something that is not joint work with Gertrand Boreau and Nicolas  Arrentain.  So they are both students of Bertrand and the geometric recursion clearly takes its  sort of origin in topological recursion, but it's quite different from topological recursion  and you will see how.  So let me just start by looking at a setup that I would like to consider.  So this is a very general setup, so I want to look at a category S, which is compact  oriented surfaces and the morphisms are just isotopic classes of diffeomorphisms.  And the target category would be just be some category of vector spaces.  You should think of infinite dimensional vector spaces, most of the applications I have in  mind.  And then I have a functor given from S to V, so it assigns a vector space to every surface  in a functorial way.  And moreover, I want a functorial assignment of vectors inside these vector spaces for  every surface.  And then of course, since it's a functorial assignment, what will happen is that these  vectors here will be in the mapping class group invariant part of these vector spaces.  And so you might think this is a very particular situation, but actually there are really a  lot of examples that are very naturally of this form in low dimensional geometry and  topology.  So let me, that's what I say right here, and let me start to give you the first few examples.  So the first example, I will simply take E of sigma to be continuous functions on Teichmann  molecular space of the surface.  Okay, that's the functor E. And then of course, I think you will certainly agree with me that  if I take the function one in every Teichmann molecular space, then that works.  Now you may think this is an utterly trivial example, but I will get back to this and it  relates very closely to Mia Sakane's work of Mia Sakane-Makshane's identities and her  work on volumes.  Okay, second example more interesting.  So let's take S to be the set of multi-curves.  So this is the set of isotopic classes embedded close one manifold in sigma such that no component  is isotopic to a boundary component nor are any two of the components isotopic.  So this is first on multi-curves.  And now I define a function on Teichmann molecular space also.  And what it does is that it sums through the set of multi-curves.  You tend to take the product over the set of components and then you apply some function  which is sufficiently fast decaying function at infinity to the length of each of the components  and then you take product or components sum over all such multi-curves.  So obviously, this is if the decay is fast enough, this will be an absolute convergent  series and it's a nice continuous function on Teichmann space and it is mapping class  could be invariant exactly because if the decay is fast enough, you can reorder the elements  here and the set here and that's all that a mapping class would be doing.  So I will be specific about this in a second.  We will actually look at boundaries and have geodesic boundaries.  But right now I'm not so specific.  I will come back to this example in great detail.  Okay, so basically this list here is just to show you that there are lots of things that  actually fit into this general scheme that I started considering.  Yes, yes, I will get to all the details of this in a second.  But for now, I just want to motivate with some overall examples.  But you will see typically we need some kind of topologies in these vector space to make  sense of these sums.  But okay, let's consider continuous functions again.  And now what I take is I take the trace that we saw this morning of F applied to minus  the directly the plus built on a differential operator on the Riemann surface where we point  in type of space.  So this gives again a continuous function provided F is decaying sufficiently fast.  Same thing with something more simple.  Suppose you take the Vapidas and Simplactic 2 form.  Well, that's a 2 form on type of space.  If you take Bayer's complex structure that fits in the framework too because the vector  space is then just smooth sections of the endomorphism bundle of the tangent bundle.  If you take closed forms on type of space, of course, this fits in the scheme as well.  You could imagine representing homology classes.  We know that we have all of these stable even dimensional classes, but we also have all  the unstable odd dimensional classes.  And we don't really know how to construct form representatives of these, right?  So that's sort of just a remark for now.  Then of course I can consider a different modular space.  I can consider the modular space of G flat connections on the surface where G is a some  semi symbol E group.  We could rather be complex or real.  And I could then take the Fogg-Rossley Poisson structure on the space.  That would be a section of the second wedge product of the tangent bundle of this modular  space.  You know what?  Rossley is visiting our center.  And one of the things he needs to do is present gain with a copy of his passport.  And this is copy from his passport.  So all right.  So the next example is the narrow seamen's chichatry complex structure on the modular  space of these flat connections.  Now of course in order to get a symplectic situation, I have to fix conjugacy classes  at all the boundary components.  And I require that the holonomia contained in those conjugacy classes.  And then I really get a nice, so this one is I think dying, but you know I get in this  case here a nice vector space which is now more complicated because it is smooth functions  from type of space to the space in which complex structures live in, namely smooth section  of the endomorphism bundle over type of space.  But the whole thing is equivariant, right?  So it's really a mapping class group invariant element in this thing, this narrow seamen's  chichatry family of complex structures.  Okay, so along the same V, I can do.  No, and it stops at this slide here, but let me just mention them.  So there is the Ricci potential in modular space.  There's the hypercalar structure of Nigel Hitchin and the Higgs bundle modular spaces.  You could even talk about representations of mapping class groups.  Itself, they will just be flat connections in trivial bundles of a type of space.  You could also talk about boundary vectors in TQFT.  They don't quite fit the scheme here because in this case here the boundary vectors are  only invariant under the diffeomorphism with the extents over the three manifold.  But okay, let's mention them anyway.  In fact, invariance of both three and four manifolds, smooth four manifolds, can be fitted  in this scheme simply by representing three and four manifolds by either HECA diagrams  or trisection diagrams.  Also forms that come out of Grumman-Witton theory and amplitudes in closed string theory  as far as you believe that they are top forms on type of space that are mapping class group  invariant.  So tons of it.  Okay, let's now be more specific then.  So the category I would like to work with is compact oriented surfaces of negative  Euler-Catzuristic with a marked point on each boundary component together with an orientation  of the boundary in such a way that it splits to in and out.  And the inclusion of the ins in the surface gives you an isomorphism on pi zeroes.  It means that there is exactly one in for each connected component and all the rest  are out.  Okay?  And then isotopic classes orientation preserving.  The amorphisms which preserves the marked points and the orientations on the boundary,  modular isotopy which preserves all the structure.  Those are the morphisms.  So this is the precise category of surfaces we want.  Okay.  So now let's go to the category of vector spaces that we want.  Well at first go what we would like to have is house of complete locally convex topological  vector spaces over the complex numbers.  So these are vector spaces whose topology is given by a semi-norms.  And then I want for morphisms, I just want morphisms of locally convex topological vector  spaces.  But we have to go a little bit more complicated than that unfortunately.  And let me just sort of preempt the example of Teichmann space.  So example of continuous functions on Teichmann spaces.  And so the point is that, you know, I'm also going to introduce precisely which Teichmann  space I want in a second.  But for now just think of Teichmann space with metrics that are hyperbolic on surface  with boundaries such as the geodesics, so the boundary of geodesics.  Then of course inside here, you see, and then I might just, so the vector space I just want  to consider is the one that we saw before that's just continuous functions on this Teichmann  space.  But in order to somehow get the whole analysis to run under the recursion, I have to consider,  you know, pick an epsilon and then consider the subspace of Teichmann space that consists  of surfaces with all closed symbol geodesics of lengths at least epsilon.  So I need those also.  And so therefore that epsilon will lie in the real line plus, and I will have C0 of  these guys.  So what happens is now that I don't have just one vector space.  I have a system of vector spaces indexed by a directed set.  So that's why I now switch to saying that an object in my target category is a directed  set and an inverse system of such house of complete locally convex topological vector  spaces over this directed set.  So here this means, you know, smallest geodesic bigger than or equal to epsilon.  So initial space is directed limit of that's right.  And therefore you get such in the directed system, so the continuous functions.  And now I also want some specific set that indexes the semi norms.  And in this case over here, you should just think of this epsilon here as well a set of  compact subsets of this system, I mean this set of system at least epsilon, right?  So then if I take the semi norms, which is max of the point wise norm over those sets,  I will get the appropriate semi norms to get the natural such vector space structure on  these two.  Okay, so that we have here, so we have these projective limits like this and then of course  what I can do, yeah I can take the projective limit of the whole guy, that's this space  here in this case here.  And then of course what I can look at is I can look at sort of the subset that has finite  norms where the norms that I'm, so semi norm where the norm is simply the supremum over  all compact subsets.  So this subspace prime here just means functions that are bounded on all the system sets.  These are epsilon system sets, right?  So the semi norms here, this guy, unfortunately this doesn't really work for that guy.  You know that's the max norm over that subset.  So not all the continuous function of course has finite norm for that, obviously, it's  only the bounded ones.  Okay, and then morphisms is completely naturally defined, the morphism of such two objects,  you know is an inverse system of continuous linear maps, phi ij, where i is running through  the first index set and j runs in the second one but it is less than h of i, h of i is  some order preserving map from one to the other and this should of course induce continuous  maps of the limits in such a way that these prime spaces are mapped to each other.  Okay, very good.  So I assume that I'm given, you know, a fountain from this category of surfaces that I introduced  to this category C here.  So such a C I call, you know, a target theory.  It will have to satisfy a few more things in a second.  But anyway, the whole idea is try to recursively define a vector inside the e prime guy of  the surface, mobbing class group invariant part and we want to do this recursing in the  Euler characteristic.  The basic idea is to remove a pair of pants from the surface and this will increase the  Euler characteristic by one.  I'm only considering negative Euler characteristic surfaces so it will end with Euler characteristic  minus one which is either a pair of pants or one whole torus.  And so over on the side here try to indicate the three different types of pair of pants  and binnings you can imagine.  Namely one is where there are two boundaries in common with the surface of the pair of  pants and the other two, so that's the B case and the other two is the C case and the C  case we only have one boundary in common and the two other boundaries are inside the surface.  Okay, now this requires me to have further data given because this means that, you know,  if I want to define something for disjoint union I would like to have a bilinear map  and morphism here from the cross product two vector spaces for the two individual components  over to the disjoint union and also if I have gluing maps, you know, if I have two surfaces  and if I have a pair of boundaries that I want to glue on I need to have gluing maps  because then I can somehow, you know, cut out this piece here each time and I can stick  in some starting data say A for a pair of pants, Mapping Class Group Invariant part of  that and D for some guy for the one whole torus also Mapping Class Group Invariant part  of that and so I will start by defining these two to be that initial data and then I need  recursion kernels so these are just elements also in the vector space for a pair of pants  both B and C and B, well B depends on a choice of one of the boundaries on the pair of pants  that namely goes to the other one, right?  So, post here this is the minus boundary and the pair of pants has to have its minus boundary  go here but then the pair of pants has two out boundaries and I can decide whether it's  one or the other that goes to this boundary here.  So therefore I have two copies of B. Okay.  All right and then the idea is simply I just sum over all possible ways of extracting pair  of pants of the two kinds of types, I stick in the recursion kernel in the first slot  and I use the morphism for gluing and I recursively assume that I have defined the thing for what  remains after I extract a pair of pants and I sum like this.  Now of course this has problems because these two sets here of isotopy classes of embedded  pair of pants these are of course infinite, I mean they're countable but they are infinite  sets.  So therefore I will have to do some kind of analysis in order to understand what this  sum means.  Now if it could actually be quite interesting to maybe just do some kind of divergent series  here where you just assume that these terms here are bounded by something for each extraction  and then you would get a divergent series and you'd have to work on this sort of the  way that we are having the main theme of this conference.  But what we will do at first is actually somehow specify conditions on these gluing maps and  the initial data in such a way that these sums here will be absolutely convergent.  Okay.  So let me now describe what we need in order to make sense of these if we want them to really  be convergent series.  And so in order to make them.  It's not that you are summing over an infinite set.  Yes I am.  Unordered infinite set.  Yes, that's right and therefore we better make sure that the series are absolutely convergent  so it doesn't matter which ordering we choose.  Right?  Yeah.  I agree.  Okay, so now what we are going to do is actually give the precise definition of what I mean  by a C-valued target theory.  So this will be a function E from S to the category C we have previously defined.  These directed sets of vector spaces with all these semi-norms on.  So that's what I'm saying here.  So I have the vector space for every surface and I have all the set of semi-norms indexed  by in this case here it was all the compact subsets of the system sets in this example.  Right?  Right now I'm just thinking abstractly there are just some abstract given set of semi-norms.  And now I take again inspiration from this example here.  So if I have a compact subset of type-molar space and I have a given curve on the surface  what I of course I can do is I can talk about the length of this curve because I can just  take the minimum of its geodesic length over this compact subset.  So we see that there is a natural way that what parameterizes the semi-norms in this  example also parameterizes length functions.  And so therefore I'm simply just going to say abstractly well I just want abstractly  length functions which are defined on the set of multi-curves on the surface.  They take value in non-zero complex numbers and they are exactly indexed by the indexing  set for the semi-norms.  So that's just a natural abstraction out of what I see in the type-molar case.  Okay.  Now I need that these length functions satisfy three different actions for this theory to  work and the first one is polynomial growth action.  And so what you do is you say let's take an i and let's take some semi-norm and what  I do is I also take a large l and then I look at all the multi-curves that has length less  than l for that set of semi-norms.  For that semi-norm sorry.  So this is a finite so what we need to assume is that this is a finite set and its size  should not grow more than polynomial in the length l.  So that means that should exist mi and di such that this guy here when I take supremum  over all the things that index the semi-norms for the i subspace should be bounded by this.  And so in the type-molar case this is simply just a result of ravine that says that honor  system set the number of geodesics of length less than epsilon sorry length and l sorry  and it's simple geodesics.  I'm talking about simple closed geodesics grow at most like a polynomial.  I think this di here is 16 minus 6 plus 2n or something like this is the best constant  you can choose in that case.  But we just assume abstractly that there is this property here for our length.  Okay then there is a lower bound and that is directly from the system definition here  we strapped this out so it says that for any i there should accept an epsilon i bigger  than zero such that the infimum of the length over the set here of all the semi-norms and  all the simple closed curves on the surface should be bigger than epsilon i sorry epsilon  i.  So that's just the analog of saying that I have fixed system sets.  Okay, then there is the small pair of pants action and it simply says that for each i  there should be some constant such that if I look at all the semi-norms guys and I look  at the length and I look at all pair of pants embedded in the surface for which the length  of the interior boundary of the embedding is smaller than the length of the boundary  that's common with sigma then this should be a finite set and bounded by Q i.  So that this is satisfied for type mirror spaces follows by some work of Hugo Parlier.  So those are the three things I would like to have polynomial growth action I would like  a lower bound action and a small pair of pants action.  So now I specify the gluing data so again I want this way to take disjoint unions so  a linear map here that satisfies nice associativity when you do Cartesian products and associativity  of unions on this side.  I would also like to have gluing maps like this when I have pairs from the two boundaries  from the out of this and in here and then I make the gluing and so I'd like to have  linear maps from the vector space of these two to the glued thing and of course this  should be compatible with gluing morphisms and with associativity of gluing and with  disjoint unions.  So this is exactly what I said before here that I was in need of in order to make sense  of this.  So very good.  So now what is the initial data that I have to start with?  Well as we discussed before I certainly need the thing for a pair of pants but I also need  the recursion kernel, the C1 and BB I need also a pair of pants where B is one of the  out boundaries of P and of course if I have a diffeomorphism that interchanges the two  out boundaries then I must have this condition here on B.  And then D and I want to assume that D is a separate guy.  I could assume always that C was some kind of trace of D because for one whole torus  I could just sum over all possible ways of breaking it into a pair of pants but that  would put some trace class conditions on C that we don't want necessary to have in all  our examples.  So I just allow D to be a separate part of the initial data.  Okay and here is the precise analytic properties, the decay action that we want for the initial  data combined with the gluing maps.  So it says that if you take two guys from the indexing set of the pair of pants and  the complement of the pair of pants for some embedding of a pair of pants inside sigma  and now you take a K that is less than this map that you get from the directed sets by  the gluing morphism and then you take any semi norm in the glued surface then you require  that there must exist an SK that is bigger than this DK that went into the growth rate  and functorial constants that depends on these indices i, j and k such that if you take  and look at the gluing morphism applied to B and any vector in the complement of the  pair of pants so that's called, where is it?  It's called sigma complement.  I get there.  So any vector V that lies in the E prime space, mapping class, gluing variant part, you must  have that this norm here is less than this constant times the norm of V times this one  plus and then you take the length of the boundary in the interior and you subtract the length  from the stuff that's common with the boundary and then you add one and you take the plus  part of that and the plus part just means if it's negative it's zero but if it's positive  it's just equal to the identity.  So you only look at the things that give you positive quantities for this quantity here  so only the pair of pants for which the length of the boundary is smaller than the interior  length and that's typically what's going to happen that's most of them in fact all  of them except for finitely many by our small pair of pants action and then you lift, you  require that this lifted to minus SK, bounce this quantity here and the same thing for  C. So this is the precise analytic condition that we require on this initial data and I'll  give you several examples in a second that satisfies this.  Okay, so now we start the recursion.  So for an empty surface we may want to define this to be just one for a pair of pants and  one whole torus we are bound to define this as A and D and now we just use this formula  here to define it for any surface of Euler-Cathoristic less than or equal to minus 2 and for connect,  this is for connected surfaces and for disconnected surfaces we just multiply them together via  this you know disjoint union morphism.  And so the main theorem of this is that this assignment is well defined and so more precisely  actually this series here is absolutely convergent for any of the semi norm and its limit lies  in the E prime guy and it is functorial.  In particular this state that we get out of it here lies in the E prime guy and it is  Moppen-Klasgrub invariant.  Okay, so let me just say a little bit about how the proof goes.  So you just for example take this part of the sum here and because we want to show absolute  convergence I just take the norm of everything inside the sum and now if I just look at,  so obviously if I just take my estimates that I have for the initial data, so I just go  back here and show you, just use this estimate here straight off.  If I use this estimate what I get is that this side here is bounded by this quantity  here where this theta function here is simply just the sum of all the pair of pants and  then you take this quantity here.  And now the axioms are exactly set up such that polynomial growth axiom plus small pair  of pants axiom will guarantee that there exists an SK bigger than DK such that this is finite.  And then the lower bound action plus the small pair of pants axiom will actually say that  there exists a constant such that this gadget here when you are taking supremum over all  the semi-norms is universally bounded.  And therefore what you are going to get is this quantity here so that shows that this  series is absolutely convergent and lies in this set here and then you have to work a  little harder to say that the whole thing lies in E prime but I don't want to go through  all the details of the proof.  So in other words you can really make sense of the sums here this way here.  Let's go through the example of Teigmuller theory and let's me introduce precisely which  Teigmuller space I want.  So I take, you know, remember that the objects I have in this category of surfaces they are  surfaces where each boundary component has a mark point.  So I denote those OB here.  And now the Teigmuller space so this guy is simply just the usual way of defining Teigmuller  space you define it to be diffeomorphism from the surface to a Riemann surface where this  is now a bordered Riemann surface, modulo and equivalence relation.  And such two guys here are equivalent if there exists a bihologomorphism from the first Riemann  surface to the second such that when you do this composition here around this first of  all restricts to the identity and the mark points and it is isotopic to the identity  on the surface via diffeomorphism which restricts the identity on these mark points so it doesn't  change these mark points.  So it isn't quite the usual Teigmuller space.  The usual Teigmuller space you wouldn't have the mark points right?  But the thing is that of course you can just forget that you have the mark points so there  is a projection from this Teigmuller space here to the usual Teigmuller space of bordered  surfaces.  And this guy here is simply an R to the number of boundary components bundled over this because  when I do the twist around the boundary you know I can keep doing it and even if I do  it too pie around I don't come back to the same metric because it's twisted right?  Okay now of course I have the group delta sigma which is all the boundary parallel dean twists  they act freely on this guy here and if I divide out that subgroup here of the Mopin  class group acting on this Teigmuller space that should have been a P here.  It's this guy here.  Then of course this just drops down to be a circle bundle over I mean a torus bundle  over this guy where the torus is this u1 to the number of boundary components.  And I need exactly this guy here because I want to integrate over these fibers here in  a second when I do the gluing morphism.  Okay so you know if we look at a pair of pants what do we have?  Well the usual Teigmuller space just says that it's R plus to the 3.  Just think of representing by hyperbolic metrics.  The boundary components are geodesics so therefore you just have the three lengths of the boundary  components for pair of pants that gives you the usual Teigmuller space.  Specifies points uniquely there.  And the point is that with these guys here you just add a copy of R for each of the boundary  components and if you divide by these dean twists that are parallel to the boundary  you just have R plus cross u1 to the 3.  And then we can of course imagine having coordinates so these are the lengths and these are the  twist parameters on this Teigmuller space here.  Okay so in a second I'll give you initial data and this means I just give you functions  that are defined here on these six variables.  In fact most of the time I'll just give you one that is defined here and then I just  put it back via this projection that forgot about the mark points.  Okay so now let's discuss how are we going to do the gluing maps.  And so let's take two guys and let's take some pairs of boundary components we're going  to glue on.  Then of course what I can look at is I can look at the subset of the Teigmuller space  of the disjoint union where I just look at the subset where the lengths of the boundaries  that I'm going to glue on agree.  So I'm ready to glue right?  So that's the subset of the Teigmuller space of disjoint union and now I can just glue  so I get a map from here to the glue to the glued surface just by gluing the hyperbolic  matrix I can do that if the lengths on the boundaries agree that I glue on.  And the point is that this will give, realize this space here where I divide by you know  if you take the, if you take the, if you have such a curve that you're gluing on if you  do a dint twist on one side and the inverse dint switch on the other side before gluing  this really matters but after gluing it doesn't because the two are of course isotopic to  the identity after you glued so therefore you can divide that group action out it won't  change the image over here and the point is that what you're left with is again just  a circle, a torus bundle over this guy here.  A copy of the circle for every boundary component, a pair of boundary components you glue on.  Okay so now let's introduce these subsets.  So sorry what I called here T epsilon sigma is actually called K sigma epsilon here.  So it is just the subset for which the system so the length of the shortest deodesic on  the surface is bigger than epsilon I take indeed just continuous functions on that.  And then of course as we just discussed over here we have a family of semi norms by just  a thinking of compact subset here that gives me this locally convex house of complete topological  vector space structure on these and of course if epsilon is smaller than epsilon prime I  get restriction maps that go this way here.  So I have a system I have a directed system as I said I would like to have and now the  vector space the total vector space we're considering as of course this continuous function all  a type molar space we look at this these norms here they are just supremum norms over the  sys-tool sets and so these guys here are just functions that are bounded on the sys-tool  sets.  Okay yeah then we define a length function and this length function is just take the  minimum of the length of a given curve over the compact subset.  Now this is actually commensurable to if you just took one point of this compact subset  because there actually exist a constant between zero and one such that you know this length  here bounds Ck times the length of the all of the elements in the compact subset.  Okay now so because this guy here is exactly defined in such a way that it is sys-tool  sets of course it satisfies the lower bound action and then as I sort of promised in the  beginning here or mentioned in the beginning right there is a result of ravine which actually  was refined by Miesa Kani but I think the way we just use it here is just the stuff  that was done by ravine that says that you know if you have a simple closed curve on  the surface or a multi-curve then if you look at the ones that have growth that have  lengths less than L then this here will grow slower than the power of L when L goes to  infinity.  So the polynomial growth action is also okay and there is this work of Hugo Parlier we  could use to show that the small pair of pants action is also okay.  Very good so now I have to give you precisely what is the gluing morphism.  Well for disjoint units are totally trivial because the tight molar space of a disjoint  unit is just a cross product so you have a projection QI to each of the factors and you  just pull back with respect to the QI's and you take the product that's the disjoint  morphism, disjoint union morphism.  Now the gluing morphism is similarly simple what you do is first you take the disjoint  union so you get tight molar space of the disjoint union then you look at this inclusion  map which is just the things where you say the ones that I want to glue on has to have  same length that's a sub-variant so you just look at the inclusion of that and you pull  back the functions to there and now you are on the circle vibration or sorry I keep saying  circle vibration I mean torus vibration because you might glue on more boundary components  than one and so you have this torus vibration and so you simply just integrate with respect  to you know the rotation invariant measure on each of the fibers of this map so that's  the gluing morphisms everything is very explicit and simple.  So initial data means that I have to give you two functions A and C of the six variables  which are symmetric in the last two variables I have to give you two B's depending on which  of the two output boundaries of the pair of hands I select to be special and so they are  just completely you know general functions there are no symmetry of those however they  are related to each other namely they if I permute the last two coordinates I go from  one to the other.  Then I must specify a continuous function that is mapping class gluing invariant on  tachymorla space of this of the of the one hole torus and now the admissibility of the  initial data simply becomes very simple in this case here it just says that for all S  and epsilon bigger than zero there must be exist a constant that depends on these two  such that if you take the supremum over the sysdel set of one plus this positive part  of the interior length minus the exterior length and you take that times the function  for the recursion kernel B this must be bounded by this constant and correspondingly for C.  Okay.  That is ABCB zero six A constraint.  No they are completely free.  That's right.  Yeah but that's because I have a single out in boundary and all the rest out.  If I want to have it the result completely symmetric in all I must satisfy relations.  Because the free theory is easier to talk about I somehow I could talk about that and we know  what the relations are in this language here.  Okay but I rather give you some examples.  So let's consider the following.  So we call this the Mieser-Khanem-Makshane initial data and so what you do is you take  A to be the constant function one and then B and C are given by these very explicit expressions  you see here and then D is just obtained by taking what I would call the partial trace  you just some C over you know here you insert the length of the boundary of the torus and  here you insert the length of the of a simple closed curve in interior that cuts this thing  into a pair of hands and then you sum of all sorts.  And so when you do this then the theorem says that for any surface the geometric recursion  applied to this initial data here with this target space continuous functions will give  you the function one.  So you know you should think of this as a kind of partition of unity okay and you know  I would say the proof of this is completely straightforward except for the one fact that  namely Mieser-Khanem's generalization of the Mieser-Khanem-Makshane identity and of course  this is why we pick these two functions they are taken directly from formally from Mieser-Khanem's  generalization of the Mieser-Khanem identity.  So the proof is simply just you iteratively each time you apply the recursion you prove  that this is the case via her identity.  Okay but the nice thing about this is that this has ramifications well sorry you will  see the ramifications second I just want to introduce another set of initial data and  this is initial data we call the conservative initial data and so A is again the constant  function one.  And now B is this gadget here so remember the plus is just the positive part right.  So you just take these very simple expressions here and for C you take this and for D you  take the sum here and so the theorem now says that you can run the geometric recursion  for these and then you will get some functions on a Teigmaler space which I'm up in classical  invariance of course there are continuous functions on the modularized space and in  fact they are integrable with respect to the volume form that there is associated to the  vapid and symplectic form and if you perform these integrals here what you get is exactly  just these evaluations here of PsyClasses and so this is the reason why we call it  the conservative initial data.  And so you didn't consider that.  So that's the same with this Mieser-Khanem and McShane didn't consider this either.  You did consider this I think.  So now I want to generalize these data.  So what I do is I look at a continuous function defined on the positive reels and it's going  to have sufficient fast decay at infinity we'll get to that but my aim is to say I want  to make this sum over simple curves I keep saying that but it's really multi-curves right?  Sum over multi-curves and then this product of f applied to the length.  And so if you define SF to be the sort of fastest I mean the polynomial decay rate of  the function so you assume that this guy here is finite or actually sorry you just defined  it this way here but if you know that you have a surface for which 6g minus 6 plus 2n  which is this growth of length of jesics that are simple and closed is less than SF then  this function here will be an absolute convergence series which defines a continuous function  on this type of space.  Okay and now for so therefore for example if you assume that SF is infinity then this  works for all surfaces and one example of that is that if you just take f to be sort  of exponential decaying then of course it's fine.  So for disconnected surfaces I just simply define the function for the disconnect serve  to be the product of the two components and I for pair of pants of course we observe that  this function is one because the only entry in this sum here is the empty multi-curve  because I don't allow anything to be isotopic to the boundary right?  Okay so now the thing is that we can just do this very simple thing of f-twesting initial  data we take the miyazakhani initial data but now we do this kind of twisting so for  b we simply just add f of l and for c we take c of miyazakhani and then we combine b with  f this way here because it has to be symmetric in the last two right so you kind of let to  this if you want to combine it with f and b and then you take the product in the end  here and so you take af to be 1 and you take df to be 1 plus the result of summing over  the symbol of those curves on the torus and so the theorem which is you know less trivial  is that when you apply the geometric recursion to this initial data here you get exactly  this function here.  So this is where geometric recursion is sort of very different from some kind of topological  field theory or something like this because if you took this function here on a surface  and you caught along one specific curve then this function will satisfy terrible complicated  gluing rules because you would want to try to have geodesics match up with directions  along the curve you caught along and so on and it's very complicated.  I don't know how to do that in fact but this geometric recursion it satisfies very nicely  and the idea of the proof is somehow that when you have this large sum you can always  fit a pair of pants inside the complement of these curves and since you're summing over  everything this is going to work out so there was a very rough schedule that proof but that's  the idea that makes it work.  Now let's suppose we have some Mappin-Klauss-Kupin variant functions on Taichmuller space and  let's further assume that they are integrable when we multiply them onto the very Peterson  volume form so I'm going to use this notation bracket 5 for the integral over the moduli  space where I fix the length L1 up to Ln which I therefore have as inputs to the function  on this side here and then if you apply this to this function here I mean it turns out  that if f for example have exponential fast decay then it automatically follows that  the function here is integrable with respect to the vape-pedeson volume form and the result  will satisfy this very simple recursion here.  So you see that this recursion here only involves the initial data that you started with B and  C twisted with F of course but then it's just integrations over R plus and R2 plus  and of course these formula here should remind you a lot of what Bertrand just talked about  topological recursion except that he didn't write down what the recursion is but I did  it here and in fact I think my next slide yes indeed is that if you look at this topological  recursion and so this topological recursion takes as Bertrand was explaining to us a  spectral curve and a one form and a two form and then you can produce out of this these  forms that are indexed by G and N and the point about how it's related to you know geometric  recursion is the following that if you start with any spectral curve which is ready to  be applied you're ready to apply topological recursion to then you look at the set of ramifications  points for X and you consider the sort of standard set of coordinates around each of the ramification  points now you take V to be the free C vector space on the ramification points then we're  saying there exists a family of admissible initial data parameterized by a parameter beta in R plus  four geometric recursion with the target theory being continuous functions on Teichmuller space  with values in V tensor power the number of components of the boundary of the surface so you  can compute the GR amplitudes we call these vectors amplitudes they become integrable over  the modular space of the guys where you fix the length of the boundaries with respect to the  usual way Peterson volume form and then what you should do is you take these averages of the guy  you constructed via geometric recursion here then you do the inverse Laplace transform in all the  variables you end up with C eyes set eyes sorry and the set eyes are exactly the coordinates here  so you take the amplitudes from topological recursion that is you know computed and you take  these iterated residues like this and you get that see beta yeah so yeah yeah okay it's in  conform field here I think might be right so yeah so this links up topological recursion with  geometric recursion in fact it's saying every time you can do something with topological  recursion you can lift this to geometric recursion you can compute amplitudes in geometric recursion  and modulo this Laplace transform this is the same thing as output okay now there is a sort of  more general thing that you can do namely anytime you have initial data for geometric  recursion where these constants that I showed you for the initial data if they are independent of  epsilon then the output of geometric recursion will be integrable over the modulized space with fixed  lengths with respect to the very Peterson volume form and if you define the WG ends of else to be  the average of these guys then this will satisfy this recursion here okay so for example if we go  back this is exactly what you saw for these functions here and so by the way this sort of  allows you to tell for answer questions like if you ask so I have formulated this for continuous  functions but there is an analog of this for measurable functions so for example if you take  f to be the function that is constant one in the interval for say from one to two and zero  otherwise then this certainly has fast decayed infinity and if you compute this what you will  be getting for these brackets here is the volume of the modulized space for which the surface has  a geodesic between one and two so answering questions about such statistics of lengths of  simple closed curves I think was open and we give this sort of nifty recursion here namely  that these satisfy topological recursion okay and so that's a very general thing as this this guy  here says this happens for all geometric recursion you can do provided you know that they are  insicable over the modulized space okay it is not only about functions so let me give you  an example where we study forms so recall that we had this type motor space for a pair of hands  was this guy here so I have six coordinates and of course if I take a torus I also have  Flenschen-Nielsen coordinates on the on that well I have to choose a curve to do it with  respect to right but then I get coordinates like this this is for the boundary and this is the  interior stuff and now I take the target theory just all forms on type motor space and I take  a to be the exponentiation of this form here so what I would like to call the way the the  the way Peterson's symplectic form for this type motor space and B you just multiply me as a  Kani initial data onto this guy here and C you just multiply me as a Kani initial data onto  this guy here D you take X of the way Peterson's symplectic form and then the theorem says that  when you run the recursion for these guys here you get indeed exponential of the way  Peterson's symplectic form on all surfaces so that's a that's an example where you're using  forms and not just continuous functions okay now we can also work on on measure of each  volume so this is some rather recent results to with these gentlemen here and and and you  know we've just put a paper in the archive a couple of weeks ago and so now you consider the  bundle of quadratic differential so what type motor space of course this guy has a natural  norm which is just take Q which Q bar and take a square root of the absolute value of that  integrated over sigma that should have been that sigma there but okay and so there are of  course local holonomic coordinates on the space of quadratic differentials and so therefore  that gives you a notion of lattice points and there is the measure of each measure on the  space of quadratic differentials which is simply obtained by counting lattice points and so we  normalize it in such the way that the co-volume of the lattice is one and then that allows you  to define a measure on the unit sphere bundle of this cotangent bundle of modular space namely  if if y is a measurable set inside the unit norm correct differentials here you simply  just defined its measure to be well 12g minus 12 plus 4n times and then the measure of each  volume of the cone on it of the size here and so this is a mapping class group invariant  measure and it's a result of measure and beats that actually the whole bundle here over modulized  space has finite volume and so it's an interesting thing to actually try to compute these measure  each volumes okay so we do the following take this function here and now take that function  and twist the conceivage initial data with this f and then what comes out we will call the measure  of each amplitudes their continuous functions on type more space which are course mapping class  group invariant and then we can they actually also satisfy that it becomes an integral function  over the modular space and so we average this function with respect to the vapedas and form  again over this modular space and it becomes a function just of the lengths of the boundaries  and our theorem says that this is a polynomial in the allies actually and this polynomial is  related to the measure of each volumes via this explicit thing here so it's zero order coefficient  is simply just modulo these quantities here the measure of each volume so in other words you  know you can construct this function via geometric recursion and then when you integrate it you get  this and so since you always have recursion when you integrate so one of the tools that geometric  recursion can be used for is to show that something is satisfying topological recursion because we  have this theorem that says that if you can find some function that satisfy geometric recursion  its average will satisfy topological I come to that in a second so far n is bigger than one  here you're equal to but I come to that on the next slide not this one but actually so if you  run through the recursion what does the recursion actually say in this case here well if you look  at this guy here was a polynomial in fact here is exactly how it is a polynomial and here are  the coefficients and so we can write down a recursion that just involves the coefficients so  you start out by setting these to be zero you put this to be this delta function and then if one  one is this guy here with this C2 of two and then you have this recursion relation here with the  B's and the C's and the B's and the C's are given explicitly like this where you see even C's  is going in in linear and quadratic way so to answer your question if the surface of genus G and  at least some boundaries the measure of each volume is given so this this is what I just said  before but it turns out actually that if you have a closed surface of genus at least two then you  can actually get the miss of each with no mark points by this coefficient here so you put L  equal to one and you look at one puncture and you're correct with this factor so this means  that you know we can very quickly you know computed via topological recursion or if you like just  using this recursion relation here we can give you all these polynomials they could be put on a  computer and spit out as high G and N you want okay so so far all my collaborators have been on  board with this and and so this so far I've just given you things we we really have to have results  and if you now allow me to speculate a little bit then the next stuff will not so if you if you  have complaints about this and if you have some I'd like to hear it but don't blame my co-authors  for it please okay so I I just want to try to make some speculations about the closed string field  theory so in closed string field theory you have a vertex Hilbert space which comes from  conformal field theory right and you have an inner product on it and the whole key about this is  that this theory provides you with brackets which is indexed by G and N and some positive real epsilon  which are you know multi linear brackets from V to the end to V and all of these brackets should  satisfy the quantum master equation now I'm not going to write down what a quantum master  equation is but just tell you that I can you know I can turn of course I can use the inner  product up here to turn these brackets into brackets like this if I like so now they're just  complex valued and I just the equivalent to these brackets here via this inner product like that I  just dualized the first variable and then the whole sort of expectation or the thing that you  or the definition of whatever you want is that these multi linear pairings are determined by  integrating certain top forms over some subsets of modular space MgN not bar here but some subset  inside the open part of the modular space and this subsets here together with the natural  properties that these forms satisfy when they come from conformal field theory is you know the  quantum master equations for these guys here is equivalent to that these subsets of modular  space is satisfies this here under of course natural identification of the forms under such  gluing so you're supposed to find some subsets somehow such that the boundary of these subsets  for Gm is obtained this recursive way here which reminds us a lot about what we've just seen and  certainly a lot about topological recursion because these are exactly the terms in topological  recursion okay so what I claim is that we can actually generate a function that had that is a  characteristic function on the modular space for a set that satisfies this analog of the quantum  master equation here and so it's very very simple you just take the following function from r plus  to r it's a measurable function it is t in the interval from 0 to epsilon and it is 0 otherwise  and I require that epsilon has to be less than ox into one now I take initial data which is just  a abcd coupled to these functions here and it is the should have been Mirza County McShane  coupled to these f's here okay and now if you for each point in teichmann space define n epsilon  sigma p to be the number of simple closed theodistics of lengths shorter than epsilon then theorem  and so that one that part is okay that's easy to check actually that if you compute the amplitudes  in geometric recursion for this function here coupled to the Mirza County McShane initial data  you will exactly get one plus t to this power here and so in other words if I take t to be  minus one the result will exactly be the indicator function of some subset and what is this subset  well this subset is simply just the systle set just the subset of modular space for which the you  know the all the simple closed interior theodistics have length at least epsilon and so if you analyze  this function here and look at its derivative in a distributional sense you will exactly see that  it's bound these sets here if you take this set here and hit it with require that all the boundaries  also have length epsilon then these sets here exactly satisfy the quantum master equation  so that of course tells you sorry that tells you that if you want to do this integral here you  just take and multiply this guy onto these forms and integrate over the whole space and so therefore  I think it's very likely that one should further be able to build the whole form here via geometric  recursion because these forms here satisfy what you expect from you know conform field theory  they satisfy factorizations and that is sort of how we prove that you know the function times the  vapidism-sumplictic form because the vapidism-sumplictic form satisfy natural factorization rules you  know we know that that we can do the integrations and we get topological recursion so I really  expect that these brackets here will satisfy topological recursion proved via geometric  recursion construction of this okay that would mean that we can compute them right so then we  should be able to compute these and we have a recursion relation on these and so on so people  who know something about string field theories would please tell me about what's what examples  needs to be done here because I think this could be doable okay another example which I think one  should really look at is a Kalabiyaw many folds so you should take M to be the modular space of  complex structures on the Kalabiyaw you take the standard line bundle that this guy has gotten by  taking the three zero part of the homology and then you're supposed to have these FGs which are the  genus G topological string amplitudes which is supposed to satisfy the BCOV holomorphic anomaly  equations like this you should take the mirror and then there should be holomorphic sections of  these line bundles which are the genus G Gromov-Witton potentials of the mirror and then the mirror  symmetry states somehow that you know FGs can be obtained from some kind of large complex  structure limit of the curly FGs and so this has been understood in genus zero but certainly not  in higher genus as far as I understand in general yeah yeah yes and so what I would like to understand  is can we expect that these FGNs these FGs here can be given by geometric recursion so I'm trying  to work on this but for now it's only just an expectation but of course then we can go on and  say well Witton really had this very nice intriguing interpretation of the BCOV equations right what he  did was he considered this set here which we also saw in Bertrand's talk a lot and the thing is that  then this set here satisfies in this case here the heat equation and so of course if you just  continue following this analog the heat equation is really a special case of parallel transport  equation with respect to the Hitchin connection where you have states in some spaces of holomorphic  sections parameterized by some space T which is parameterizing complex structure some pre-quantizable  symplectic manifold compact with a line model over it and so I would say that one should expect  that this parallel transport problem here can be solved in this form here again where these FGs  here should maybe be constructable via geometric recursion maybe one should first go and restrict  to the case where this M here is the modular space of flat SUM connections on the surface so I don't  know but I would like to discuss this with people if they're interested I can say the following  or let me be more precise if you like oops sorry take X to be a compact oriental three manifold  boundary sigma consider the modular space of flat SUM connections on sigma take the  transimons line bundle take the complement of the zero section of the dual line bundle then you  can embed all the sections of L to the K in one and the same space namely functions on that total  space that transforms in the case guy for the case power therefore in that space it actually  makes sense to sort of consider the total quantum transimons state series like this which is now  holomorphic function on the total space of this line bundle here of course all of it  parameterized by Teichmuller space and so the conjecture would say that the total state series  are resurgent and it has its singularities contained in the set of exponentiated transimons  values and the zero section inside the pullback of this line bundle to the SLNC modulized space  via the cotangent vibration using non-abelian hodge and then you should look at modulized  space of flat SLN connections on X and compute their transimons values and so this is should  be where this guy has its singularities so I think tomorrow we will see maybe an analog of this  in the case where the surface is empty namely for compact manifolds that Starwars might talk about  but I don't know anyway all I can say is that actually this conjecture here really works if  you look at the complement of Toros knots in S3 and that's actually related to some work that I  have done very recently with my PhD student William Miskard and there we established for  cypher fiber manifolds that the Otsuki series is resurgent and that the singularity set is a subset  of the classical SL2C transimons values and that part is on the archive this part here so this  this guy came out some four months ago William or one yeah so yeah William is here you can talk to  him about it Starwars just told me I have to finish very quickly higher teichmuller theory  could probably also be used if you want to get at the Selberg trace formula via this I don't think  you can cut along pair of hands for the same reasons what you should do is you should develop a  theory where you extract triangles instead of pair of hands then you will have corners on the  boundaries and this will be kind of an open geometric recursion we're working on developing  this and I would think that we can prove after we've done that that the Selberg trace fits in  that scheme and we can see that that would satisfy open geometric recursion therefore get same kinds  of results on averages and such things thank you very much  the problem is constructing a steady structure  I would like to hear more details about that in your joint paper that's out already I mean  yes okay but you say it you don't think it satisfies yeah questions
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This is joint work, most of it, from the start on. I will say when I get to something that is not joint work with Gertrand Boreau and Nicolas Arrentain. So they are both students of Bertrand and the geometric recursion clearly takes its sort of origin in topological recursion, but it's quite different from topological recursion and you will see how. So let me just start by looking at a setup that I would like to consider. So this is a very general setup, so I want to look at a category S, which is compact oriented surfaces and the morphisms are just isotopic classes of diffeomorphisms. And the target category would be just be some category of vector spaces. You should think of infinite dimensional vector spaces, most of the applications I have in mind. And then I have a functor given from S to V, so it assigns a vector space to every surface in a functorial way. And moreover, I want a functorial assignment of vectors inside these vector spaces for every surface. And then of course, since it's a functorial assignment, what will happen is that these vectors here will be in the mapping class group invariant part of these vector spaces. And so you might think this is a very particular situation, but actually there are really a lot of examples that are very naturally of this form in low dimensional geometry and topology. So let me, that's what I say right here, and let me start to give you the first few examples. So the first example, I will simply take E of sigma to be continuous functions on Teichmann molecular space of the surface. Okay, that's the functor E. And then of course, I think you will certainly agree with me that if I take the function one in every Teichmann molecular space, then that works. Now you may think this is an utterly trivial example, but I will get back to this and it relates very closely to Mia Sakane's work of Mia Sakane-Makshane's identities and her work on volumes. Okay, second example more interesting. So let's take S to be the set of multi-curves. So this is the set of isotopic classes embedded close one manifold in sigma such that no component is isotopic to a boundary component nor are any two of the components isotopic. So this is first on multi-curves. And now I define a function on Teichmann molecular space also. And what it does is that it sums through the set of multi-curves. You tend to take the product over the set of components and then you apply some function which is sufficiently fast decaying function at infinity to the length of each of the components and then you take product or components sum over all such multi-curves. So obviously, this is if the decay is fast enough, this will be an absolute convergent series and it's a nice continuous function on Teichmann space and it is mapping class could be invariant exactly because if the decay is fast enough, you can reorder the elements here and the set here and that's all that a mapping class would be doing. So I will be specific about this in a second. We will actually look at boundaries and have geodesic boundaries. But right now I'm not so specific. I will come back to this example in great detail. Okay, so basically this list here is just to show you that there are lots of things that actually fit into this general scheme that I started considering. Yes, yes, I will get to all the details of this in a second. But for now, I just want to motivate with some overall examples. But you will see typically we need some kind of topologies in these vector space to make sense of these sums. But okay, let's consider continuous functions again. And now what I take is I take the trace that we saw this morning of F applied to minus the directly the plus built on a differential operator on the Riemann surface where we point in type of space. So this gives again a continuous function provided F is decaying sufficiently fast. Same thing with something more simple. Suppose you take the Vapidas and Simplactic 2 form. Well, that's a 2 form on type of space. If you take Bayer's complex structure that fits in the framework too because the vector space is then just smooth sections of the endomorphism bundle of the tangent bundle. If you take closed forms on type of space, of course, this fits in the scheme as well. You could imagine representing homology classes. We know that we have all of these stable even dimensional classes, but we also have all the unstable odd dimensional classes. And we don't really know how to construct form representatives of these, right? So that's sort of just a remark for now. Then of course I can consider a different modular space. I can consider the modular space of G flat connections on the surface where G is a some semi symbol E group. We could rather be complex or real. And I could then take the Fogg-Rossley Poisson structure on the space. That would be a section of the second wedge product of the tangent bundle of this modular space. You know what? Rossley is visiting our center. And one of the things he needs to do is present gain with a copy of his passport. And this is copy from his passport. So all right. So the next example is the narrow seamen's chichatry complex structure on the modular space of these flat connections. Now of course in order to get a symplectic situation, I have to fix conjugacy classes at all the boundary components. And I require that the holonomia contained in those conjugacy classes. And then I really get a nice, so this one is I think dying, but you know I get in this case here a nice vector space which is now more complicated because it is smooth functions from type of space to the space in which complex structures live in, namely smooth section of the endomorphism bundle over type of space. But the whole thing is equivariant, right? So it's really a mapping class group invariant element in this thing, this narrow seamen's chichatry family of complex structures. Okay, so along the same V, I can do. No, and it stops at this slide here, but let me just mention them. So there is the Ricci potential in modular space. There's the hypercalar structure of Nigel Hitchin and the Higgs bundle modular spaces. You could even talk about representations of mapping class groups. Itself, they will just be flat connections in trivial bundles of a type of space. You could also talk about boundary vectors in TQFT. They don't quite fit the scheme here because in this case here the boundary vectors are only invariant under the diffeomorphism with the extents over the three manifold. But okay, let's mention them anyway. In fact, invariance of both three and four manifolds, smooth four manifolds, can be fitted in this scheme simply by representing three and four manifolds by either HECA diagrams or trisection diagrams. Also forms that come out of Grumman-Witton theory and amplitudes in closed string theory as far as you believe that they are top forms on type of space that are mapping class group invariant. So tons of it. Okay, let's now be more specific then. So the category I would like to work with is compact oriented surfaces of negative Euler-Catzuristic with a marked point on each boundary component together with an orientation of the boundary in such a way that it splits to in and out. And the inclusion of the ins in the surface gives you an isomorphism on pi zeroes. It means that there is exactly one in for each connected component and all the rest are out. Okay? And then isotopic classes orientation preserving. The amorphisms which preserves the marked points and the orientations on the boundary, modular isotopy which preserves all the structure. Those are the morphisms. So this is the precise category of surfaces we want. Okay. So now let's go to the category of vector spaces that we want. Well at first go what we would like to have is house of complete locally convex topological vector spaces over the complex numbers. So these are vector spaces whose topology is given by a semi-norms. And then I want for morphisms, I just want morphisms of locally convex topological vector spaces. But we have to go a little bit more complicated than that unfortunately. And let me just sort of preempt the example of Teichmann space. So example of continuous functions on Teichmann spaces. And so the point is that, you know, I'm also going to introduce precisely which Teichmann space I want in a second. But for now just think of Teichmann space with metrics that are hyperbolic on surface with boundaries such as the geodesics, so the boundary of geodesics. Then of course inside here, you see, and then I might just, so the vector space I just want to consider is the one that we saw before that's just continuous functions on this Teichmann space. But in order to somehow get the whole analysis to run under the recursion, I have to consider, you know, pick an epsilon and then consider the subspace of Teichmann space that consists of surfaces with all closed symbol geodesics of lengths at least epsilon. So I need those also. And so therefore that epsilon will lie in the real line plus, and I will have C0 of these guys. So what happens is now that I don't have just one vector space. I have a system of vector spaces indexed by a directed set. So that's why I now switch to saying that an object in my target category is a directed set and an inverse system of such house of complete locally convex topological vector spaces over this directed set. So here this means, you know, smallest geodesic bigger than or equal to epsilon. So initial space is directed limit of that's right. And therefore you get such in the directed system, so the continuous functions. And now I also want some specific set that indexes the semi norms. And in this case over here, you should just think of this epsilon here as well a set of compact subsets of this system, I mean this set of system at least epsilon, right? So then if I take the semi norms, which is max of the point wise norm over those sets, I will get the appropriate semi norms to get the natural such vector space structure on these two. Okay, so that we have here, so we have these projective limits like this and then of course what I can do, yeah I can take the projective limit of the whole guy, that's this space here in this case here. And then of course what I can look at is I can look at sort of the subset that has finite norms where the norms that I'm, so semi norm where the norm is simply the supremum over all compact subsets. So this subspace prime here just means functions that are bounded on all the system sets. These are epsilon system sets, right? So the semi norms here, this guy, unfortunately this doesn't really work for that guy. You know that's the max norm over that subset. So not all the continuous function of course has finite norm for that, obviously, it's only the bounded ones. Okay, and then morphisms is completely naturally defined, the morphism of such two objects, you know is an inverse system of continuous linear maps, phi ij, where i is running through the first index set and j runs in the second one but it is less than h of i, h of i is some order preserving map from one to the other and this should of course induce continuous maps of the limits in such a way that these prime spaces are mapped to each other. Okay, very good. So I assume that I'm given, you know, a fountain from this category of surfaces that I introduced to this category C here. So such a C I call, you know, a target theory. It will have to satisfy a few more things in a second. But anyway, the whole idea is try to recursively define a vector inside the e prime guy of the surface, mobbing class group invariant part and we want to do this recursing in the Euler characteristic. The basic idea is to remove a pair of pants from the surface and this will increase the Euler characteristic by one. I'm only considering negative Euler characteristic surfaces so it will end with Euler characteristic minus one which is either a pair of pants or one whole torus. And so over on the side here try to indicate the three different types of pair of pants and binnings you can imagine. Namely one is where there are two boundaries in common with the surface of the pair of pants and the other two, so that's the B case and the other two is the C case and the C case we only have one boundary in common and the two other boundaries are inside the surface. Okay, now this requires me to have further data given because this means that, you know, if I want to define something for disjoint union I would like to have a bilinear map and morphism here from the cross product two vector spaces for the two individual components over to the disjoint union and also if I have gluing maps, you know, if I have two surfaces and if I have a pair of boundaries that I want to glue on I need to have gluing maps because then I can somehow, you know, cut out this piece here each time and I can stick in some starting data say A for a pair of pants, Mapping Class Group Invariant part of that and D for some guy for the one whole torus also Mapping Class Group Invariant part of that and so I will start by defining these two to be that initial data and then I need recursion kernels so these are just elements also in the vector space for a pair of pants both B and C and B, well B depends on a choice of one of the boundaries on the pair of pants that namely goes to the other one, right? So, post here this is the minus boundary and the pair of pants has to have its minus boundary go here but then the pair of pants has two out boundaries and I can decide whether it's one or the other that goes to this boundary here. So therefore I have two copies of B. Okay. All right and then the idea is simply I just sum over all possible ways of extracting pair of pants of the two kinds of types, I stick in the recursion kernel in the first slot and I use the morphism for gluing and I recursively assume that I have defined the thing for what remains after I extract a pair of pants and I sum like this. Now of course this has problems because these two sets here of isotopy classes of embedded pair of pants these are of course infinite, I mean they're countable but they are infinite sets. So therefore I will have to do some kind of analysis in order to understand what this sum means. Now if it could actually be quite interesting to maybe just do some kind of divergent series here where you just assume that these terms here are bounded by something for each extraction and then you would get a divergent series and you'd have to work on this sort of the way that we are having the main theme of this conference. But what we will do at first is actually somehow specify conditions on these gluing maps and the initial data in such a way that these sums here will be absolutely convergent. Okay. So let me now describe what we need in order to make sense of these if we want them to really be convergent series. And so in order to make them. It's not that you are summing over an infinite set. Yes I am. Unordered infinite set. Yes, that's right and therefore we better make sure that the series are absolutely convergent so it doesn't matter which ordering we choose. Right? Yeah. I agree. Okay, so now what we are going to do is actually give the precise definition of what I mean by a C-valued target theory. So this will be a function E from S to the category C we have previously defined. These directed sets of vector spaces with all these semi-norms on. So that's what I'm saying here. So I have the vector space for every surface and I have all the set of semi-norms indexed by in this case here it was all the compact subsets of the system sets in this example. Right? Right now I'm just thinking abstractly there are just some abstract given set of semi-norms. And now I take again inspiration from this example here. So if I have a compact subset of type-molar space and I have a given curve on the surface what I of course I can do is I can talk about the length of this curve because I can just take the minimum of its geodesic length over this compact subset. So we see that there is a natural way that what parameterizes the semi-norms in this example also parameterizes length functions. And so therefore I'm simply just going to say abstractly well I just want abstractly length functions which are defined on the set of multi-curves on the surface. They take value in non-zero complex numbers and they are exactly indexed by the indexing set for the semi-norms. So that's just a natural abstraction out of what I see in the type-molar case. Okay. Now I need that these length functions satisfy three different actions for this theory to work and the first one is polynomial growth action. And so what you do is you say let's take an i and let's take some semi-norm and what I do is I also take a large l and then I look at all the multi-curves that has length less than l for that set of semi-norms. For that semi-norm sorry. So this is a finite so what we need to assume is that this is a finite set and its size should not grow more than polynomial in the length l. So that means that should exist mi and di such that this guy here when I take supremum over all the things that index the semi-norms for the i subspace should be bounded by this. And so in the type-molar case this is simply just a result of ravine that says that honor system set the number of geodesics of length less than epsilon sorry length and l sorry and it's simple geodesics. I'm talking about simple closed geodesics grow at most like a polynomial. I think this di here is 16 minus 6 plus 2n or something like this is the best constant you can choose in that case. But we just assume abstractly that there is this property here for our length. Okay then there is a lower bound and that is directly from the system definition here we strapped this out so it says that for any i there should accept an epsilon i bigger than zero such that the infimum of the length over the set here of all the semi-norms and all the simple closed curves on the surface should be bigger than epsilon i sorry epsilon i. So that's just the analog of saying that I have fixed system sets. Okay, then there is the small pair of pants action and it simply says that for each i there should be some constant such that if I look at all the semi-norms guys and I look at the length and I look at all pair of pants embedded in the surface for which the length of the interior boundary of the embedding is smaller than the length of the boundary that's common with sigma then this should be a finite set and bounded by Q i. So that this is satisfied for type mirror spaces follows by some work of Hugo Parlier. So those are the three things I would like to have polynomial growth action I would like a lower bound action and a small pair of pants action. So now I specify the gluing data so again I want this way to take disjoint unions so a linear map here that satisfies nice associativity when you do Cartesian products and associativity of unions on this side. I would also like to have gluing maps like this when I have pairs from the two boundaries from the out of this and in here and then I make the gluing and so I'd like to have linear maps from the vector space of these two to the glued thing and of course this should be compatible with gluing morphisms and with associativity of gluing and with disjoint unions. So this is exactly what I said before here that I was in need of in order to make sense of this. So very good. So now what is the initial data that I have to start with? Well as we discussed before I certainly need the thing for a pair of pants but I also need the recursion kernel, the C1 and BB I need also a pair of pants where B is one of the out boundaries of P and of course if I have a diffeomorphism that interchanges the two out boundaries then I must have this condition here on B. And then D and I want to assume that D is a separate guy. I could assume always that C was some kind of trace of D because for one whole torus I could just sum over all possible ways of breaking it into a pair of pants but that would put some trace class conditions on C that we don't want necessary to have in all our examples. So I just allow D to be a separate part of the initial data. Okay and here is the precise analytic properties, the decay action that we want for the initial data combined with the gluing maps. So it says that if you take two guys from the indexing set of the pair of pants and the complement of the pair of pants for some embedding of a pair of pants inside sigma and now you take a K that is less than this map that you get from the directed sets by the gluing morphism and then you take any semi norm in the glued surface then you require that there must exist an SK that is bigger than this DK that went into the growth rate and functorial constants that depends on these indices i, j and k such that if you take and look at the gluing morphism applied to B and any vector in the complement of the pair of pants so that's called, where is it? It's called sigma complement. I get there. So any vector V that lies in the E prime space, mapping class, gluing variant part, you must have that this norm here is less than this constant times the norm of V times this one plus and then you take the length of the boundary in the interior and you subtract the length from the stuff that's common with the boundary and then you add one and you take the plus part of that and the plus part just means if it's negative it's zero but if it's positive it's just equal to the identity. So you only look at the things that give you positive quantities for this quantity here so only the pair of pants for which the length of the boundary is smaller than the interior length and that's typically what's going to happen that's most of them in fact all of them except for finitely many by our small pair of pants action and then you lift, you require that this lifted to minus SK, bounce this quantity here and the same thing for C. So this is the precise analytic condition that we require on this initial data and I'll give you several examples in a second that satisfies this. Okay, so now we start the recursion. So for an empty surface we may want to define this to be just one for a pair of pants and one whole torus we are bound to define this as A and D and now we just use this formula here to define it for any surface of Euler-Cathoristic less than or equal to minus 2 and for connect, this is for connected surfaces and for disconnected surfaces we just multiply them together via this you know disjoint union morphism. And so the main theorem of this is that this assignment is well defined and so more precisely actually this series here is absolutely convergent for any of the semi norm and its limit lies in the E prime guy and it is functorial. In particular this state that we get out of it here lies in the E prime guy and it is Moppen-Klasgrub invariant. Okay, so let me just say a little bit about how the proof goes. So you just for example take this part of the sum here and because we want to show absolute convergence I just take the norm of everything inside the sum and now if I just look at, so obviously if I just take my estimates that I have for the initial data, so I just go back here and show you, just use this estimate here straight off. If I use this estimate what I get is that this side here is bounded by this quantity here where this theta function here is simply just the sum of all the pair of pants and then you take this quantity here. And now the axioms are exactly set up such that polynomial growth axiom plus small pair of pants axiom will guarantee that there exists an SK bigger than DK such that this is finite. And then the lower bound action plus the small pair of pants axiom will actually say that there exists a constant such that this gadget here when you are taking supremum over all the semi-norms is universally bounded. And therefore what you are going to get is this quantity here so that shows that this series is absolutely convergent and lies in this set here and then you have to work a little harder to say that the whole thing lies in E prime but I don't want to go through all the details of the proof. So in other words you can really make sense of the sums here this way here. Let's go through the example of Teigmuller theory and let's me introduce precisely which Teigmuller space I want. So I take, you know, remember that the objects I have in this category of surfaces they are surfaces where each boundary component has a mark point. So I denote those OB here. And now the Teigmuller space so this guy is simply just the usual way of defining Teigmuller space you define it to be diffeomorphism from the surface to a Riemann surface where this is now a bordered Riemann surface, modulo and equivalence relation. And such two guys here are equivalent if there exists a bihologomorphism from the first Riemann surface to the second such that when you do this composition here around this first of all restricts to the identity and the mark points and it is isotopic to the identity on the surface via diffeomorphism which restricts the identity on these mark points so it doesn't change these mark points. So it isn't quite the usual Teigmuller space. The usual Teigmuller space you wouldn't have the mark points right? But the thing is that of course you can just forget that you have the mark points so there is a projection from this Teigmuller space here to the usual Teigmuller space of bordered surfaces. And this guy here is simply an R to the number of boundary components bundled over this because when I do the twist around the boundary you know I can keep doing it and even if I do it too pie around I don't come back to the same metric because it's twisted right? Okay now of course I have the group delta sigma which is all the boundary parallel dean twists they act freely on this guy here and if I divide out that subgroup here of the Mopin class group acting on this Teigmuller space that should have been a P here. It's this guy here. Then of course this just drops down to be a circle bundle over I mean a torus bundle over this guy where the torus is this u1 to the number of boundary components. And I need exactly this guy here because I want to integrate over these fibers here in a second when I do the gluing morphism. Okay so you know if we look at a pair of pants what do we have? Well the usual Teigmuller space just says that it's R plus to the 3. Just think of representing by hyperbolic metrics. The boundary components are geodesics so therefore you just have the three lengths of the boundary components for pair of pants that gives you the usual Teigmuller space. Specifies points uniquely there. And the point is that with these guys here you just add a copy of R for each of the boundary components and if you divide by these dean twists that are parallel to the boundary you just have R plus cross u1 to the 3. And then we can of course imagine having coordinates so these are the lengths and these are the twist parameters on this Teigmuller space here. Okay so in a second I'll give you initial data and this means I just give you functions that are defined here on these six variables. In fact most of the time I'll just give you one that is defined here and then I just put it back via this projection that forgot about the mark points. Okay so now let's discuss how are we going to do the gluing maps. And so let's take two guys and let's take some pairs of boundary components we're going to glue on. Then of course what I can look at is I can look at the subset of the Teigmuller space of the disjoint union where I just look at the subset where the lengths of the boundaries that I'm going to glue on agree. So I'm ready to glue right? So that's the subset of the Teigmuller space of disjoint union and now I can just glue so I get a map from here to the glue to the glued surface just by gluing the hyperbolic matrix I can do that if the lengths on the boundaries agree that I glue on. And the point is that this will give, realize this space here where I divide by you know if you take the, if you take the, if you have such a curve that you're gluing on if you do a dint twist on one side and the inverse dint switch on the other side before gluing this really matters but after gluing it doesn't because the two are of course isotopic to the identity after you glued so therefore you can divide that group action out it won't change the image over here and the point is that what you're left with is again just a circle, a torus bundle over this guy here. A copy of the circle for every boundary component, a pair of boundary components you glue on. Okay so now let's introduce these subsets. So sorry what I called here T epsilon sigma is actually called K sigma epsilon here. So it is just the subset for which the system so the length of the shortest deodesic on the surface is bigger than epsilon I take indeed just continuous functions on that. And then of course as we just discussed over here we have a family of semi norms by just a thinking of compact subset here that gives me this locally convex house of complete topological vector space structure on these and of course if epsilon is smaller than epsilon prime I get restriction maps that go this way here. So I have a system I have a directed system as I said I would like to have and now the vector space the total vector space we're considering as of course this continuous function all a type molar space we look at this these norms here they are just supremum norms over the sys-tool sets and so these guys here are just functions that are bounded on the sys-tool sets. Okay yeah then we define a length function and this length function is just take the minimum of the length of a given curve over the compact subset. Now this is actually commensurable to if you just took one point of this compact subset because there actually exist a constant between zero and one such that you know this length here bounds Ck times the length of the all of the elements in the compact subset. Okay now so because this guy here is exactly defined in such a way that it is sys-tool sets of course it satisfies the lower bound action and then as I sort of promised in the beginning here or mentioned in the beginning right there is a result of ravine which actually was refined by Miesa Kani but I think the way we just use it here is just the stuff that was done by ravine that says that you know if you have a simple closed curve on the surface or a multi-curve then if you look at the ones that have growth that have lengths less than L then this here will grow slower than the power of L when L goes to infinity. So the polynomial growth action is also okay and there is this work of Hugo Parlier we could use to show that the small pair of pants action is also okay. Very good so now I have to give you precisely what is the gluing morphism. Well for disjoint units are totally trivial because the tight molar space of a disjoint unit is just a cross product so you have a projection QI to each of the factors and you just pull back with respect to the QI's and you take the product that's the disjoint morphism, disjoint union morphism. Now the gluing morphism is similarly simple what you do is first you take the disjoint union so you get tight molar space of the disjoint union then you look at this inclusion map which is just the things where you say the ones that I want to glue on has to have same length that's a sub-variant so you just look at the inclusion of that and you pull back the functions to there and now you are on the circle vibration or sorry I keep saying circle vibration I mean torus vibration because you might glue on more boundary components than one and so you have this torus vibration and so you simply just integrate with respect to you know the rotation invariant measure on each of the fibers of this map so that's the gluing morphisms everything is very explicit and simple. So initial data means that I have to give you two functions A and C of the six variables which are symmetric in the last two variables I have to give you two B's depending on which of the two output boundaries of the pair of hands I select to be special and so they are just completely you know general functions there are no symmetry of those however they are related to each other namely they if I permute the last two coordinates I go from one to the other. Then I must specify a continuous function that is mapping class gluing invariant on tachymorla space of this of the of the one hole torus and now the admissibility of the initial data simply becomes very simple in this case here it just says that for all S and epsilon bigger than zero there must be exist a constant that depends on these two such that if you take the supremum over the sysdel set of one plus this positive part of the interior length minus the exterior length and you take that times the function for the recursion kernel B this must be bounded by this constant and correspondingly for C. Okay. That is ABCB zero six A constraint. No they are completely free. That's right. Yeah but that's because I have a single out in boundary and all the rest out. If I want to have it the result completely symmetric in all I must satisfy relations. Because the free theory is easier to talk about I somehow I could talk about that and we know what the relations are in this language here. Okay but I rather give you some examples. So let's consider the following. So we call this the Mieser-Khanem-Makshane initial data and so what you do is you take A to be the constant function one and then B and C are given by these very explicit expressions you see here and then D is just obtained by taking what I would call the partial trace you just some C over you know here you insert the length of the boundary of the torus and here you insert the length of the of a simple closed curve in interior that cuts this thing into a pair of hands and then you sum of all sorts. And so when you do this then the theorem says that for any surface the geometric recursion applied to this initial data here with this target space continuous functions will give you the function one. So you know you should think of this as a kind of partition of unity okay and you know I would say the proof of this is completely straightforward except for the one fact that namely Mieser-Khanem's generalization of the Mieser-Khanem-Makshane identity and of course this is why we pick these two functions they are taken directly from formally from Mieser-Khanem's generalization of the Mieser-Khanem identity. So the proof is simply just you iteratively each time you apply the recursion you prove that this is the case via her identity. Okay but the nice thing about this is that this has ramifications well sorry you will see the ramifications second I just want to introduce another set of initial data and this is initial data we call the conservative initial data and so A is again the constant function one. And now B is this gadget here so remember the plus is just the positive part right. So you just take these very simple expressions here and for C you take this and for D you take the sum here and so the theorem now says that you can run the geometric recursion for these and then you will get some functions on a Teigmaler space which I'm up in classical invariance of course there are continuous functions on the modularized space and in fact they are integrable with respect to the volume form that there is associated to the vapid and symplectic form and if you perform these integrals here what you get is exactly just these evaluations here of PsyClasses and so this is the reason why we call it the conservative initial data. And so you didn't consider that. So that's the same with this Mieser-Khanem and McShane didn't consider this either. You did consider this I think. So now I want to generalize these data. So what I do is I look at a continuous function defined on the positive reels and it's going to have sufficient fast decay at infinity we'll get to that but my aim is to say I want to make this sum over simple curves I keep saying that but it's really multi-curves right? Sum over multi-curves and then this product of f applied to the length. And so if you define SF to be the sort of fastest I mean the polynomial decay rate of the function so you assume that this guy here is finite or actually sorry you just defined it this way here but if you know that you have a surface for which 6g minus 6 plus 2n which is this growth of length of jesics that are simple and closed is less than SF then this function here will be an absolute convergence series which defines a continuous function on this type of space. Okay and now for so therefore for example if you assume that SF is infinity then this works for all surfaces and one example of that is that if you just take f to be sort of exponential decaying then of course it's fine. So for disconnected surfaces I just simply define the function for the disconnect serve to be the product of the two components and I for pair of pants of course we observe that this function is one because the only entry in this sum here is the empty multi-curve because I don't allow anything to be isotopic to the boundary right? Okay so now the thing is that we can just do this very simple thing of f-twesting initial data we take the miyazakhani initial data but now we do this kind of twisting so for b we simply just add f of l and for c we take c of miyazakhani and then we combine b with f this way here because it has to be symmetric in the last two right so you kind of let to this if you want to combine it with f and b and then you take the product in the end here and so you take af to be 1 and you take df to be 1 plus the result of summing over the symbol of those curves on the torus and so the theorem which is you know less trivial is that when you apply the geometric recursion to this initial data here you get exactly this function here. So this is where geometric recursion is sort of very different from some kind of topological field theory or something like this because if you took this function here on a surface and you caught along one specific curve then this function will satisfy terrible complicated gluing rules because you would want to try to have geodesics match up with directions along the curve you caught along and so on and it's very complicated. I don't know how to do that in fact but this geometric recursion it satisfies very nicely and the idea of the proof is somehow that when you have this large sum you can always fit a pair of pants inside the complement of these curves and since you're summing over everything this is going to work out so there was a very rough schedule that proof but that's the idea that makes it work. Now let's suppose we have some Mappin-Klauss-Kupin variant functions on Taichmuller space and let's further assume that they are integrable when we multiply them onto the very Peterson volume form so I'm going to use this notation bracket 5 for the integral over the moduli space where I fix the length L1 up to Ln which I therefore have as inputs to the function on this side here and then if you apply this to this function here I mean it turns out that if f for example have exponential fast decay then it automatically follows that the function here is integrable with respect to the vape-pedeson volume form and the result will satisfy this very simple recursion here. So you see that this recursion here only involves the initial data that you started with B and C twisted with F of course but then it's just integrations over R plus and R2 plus and of course these formula here should remind you a lot of what Bertrand just talked about topological recursion except that he didn't write down what the recursion is but I did it here and in fact I think my next slide yes indeed is that if you look at this topological recursion and so this topological recursion takes as Bertrand was explaining to us a spectral curve and a one form and a two form and then you can produce out of this these forms that are indexed by G and N and the point about how it's related to you know geometric recursion is the following that if you start with any spectral curve which is ready to be applied you're ready to apply topological recursion to then you look at the set of ramifications points for X and you consider the sort of standard set of coordinates around each of the ramification points now you take V to be the free C vector space on the ramification points then we're saying there exists a family of admissible initial data parameterized by a parameter beta in R plus four geometric recursion with the target theory being continuous functions on Teichmuller space with values in V tensor power the number of components of the boundary of the surface so you can compute the GR amplitudes we call these vectors amplitudes they become integrable over the modular space of the guys where you fix the length of the boundaries with respect to the usual way Peterson volume form and then what you should do is you take these averages of the guy you constructed via geometric recursion here then you do the inverse Laplace transform in all the variables you end up with C eyes set eyes sorry and the set eyes are exactly the coordinates here so you take the amplitudes from topological recursion that is you know computed and you take these iterated residues like this and you get that see beta yeah so yeah yeah okay it's in conform field here I think might be right so yeah so this links up topological recursion with geometric recursion in fact it's saying every time you can do something with topological recursion you can lift this to geometric recursion you can compute amplitudes in geometric recursion and modulo this Laplace transform this is the same thing as output okay now there is a sort of more general thing that you can do namely anytime you have initial data for geometric recursion where these constants that I showed you for the initial data if they are independent of epsilon then the output of geometric recursion will be integrable over the modulized space with fixed lengths with respect to the very Peterson volume form and if you define the WG ends of else to be the average of these guys then this will satisfy this recursion here okay so for example if we go back this is exactly what you saw for these functions here and so by the way this sort of allows you to tell for answer questions like if you ask so I have formulated this for continuous functions but there is an analog of this for measurable functions so for example if you take f to be the function that is constant one in the interval for say from one to two and zero otherwise then this certainly has fast decayed infinity and if you compute this what you will be getting for these brackets here is the volume of the modulized space for which the surface has a geodesic between one and two so answering questions about such statistics of lengths of simple closed curves I think was open and we give this sort of nifty recursion here namely that these satisfy topological recursion okay and so that's a very general thing as this this guy here says this happens for all geometric recursion you can do provided you know that they are insicable over the modulized space okay it is not only about functions so let me give you an example where we study forms so recall that we had this type motor space for a pair of hands was this guy here so I have six coordinates and of course if I take a torus I also have Flenschen-Nielsen coordinates on the on that well I have to choose a curve to do it with respect to right but then I get coordinates like this this is for the boundary and this is the interior stuff and now I take the target theory just all forms on type motor space and I take a to be the exponentiation of this form here so what I would like to call the way the the the way Peterson's symplectic form for this type motor space and B you just multiply me as a Kani initial data onto this guy here and C you just multiply me as a Kani initial data onto this guy here D you take X of the way Peterson's symplectic form and then the theorem says that when you run the recursion for these guys here you get indeed exponential of the way Peterson's symplectic form on all surfaces so that's a that's an example where you're using forms and not just continuous functions okay now we can also work on on measure of each volume so this is some rather recent results to with these gentlemen here and and and you know we've just put a paper in the archive a couple of weeks ago and so now you consider the bundle of quadratic differential so what type motor space of course this guy has a natural norm which is just take Q which Q bar and take a square root of the absolute value of that integrated over sigma that should have been that sigma there but okay and so there are of course local holonomic coordinates on the space of quadratic differentials and so therefore that gives you a notion of lattice points and there is the measure of each measure on the space of quadratic differentials which is simply obtained by counting lattice points and so we normalize it in such the way that the co-volume of the lattice is one and then that allows you to define a measure on the unit sphere bundle of this cotangent bundle of modular space namely if if y is a measurable set inside the unit norm correct differentials here you simply just defined its measure to be well 12g minus 12 plus 4n times and then the measure of each volume of the cone on it of the size here and so this is a mapping class group invariant measure and it's a result of measure and beats that actually the whole bundle here over modulized space has finite volume and so it's an interesting thing to actually try to compute these measure each volumes okay so we do the following take this function here and now take that function and twist the conceivage initial data with this f and then what comes out we will call the measure of each amplitudes their continuous functions on type more space which are course mapping class group invariant and then we can they actually also satisfy that it becomes an integral function over the modular space and so we average this function with respect to the vapedas and form again over this modular space and it becomes a function just of the lengths of the boundaries and our theorem says that this is a polynomial in the allies actually and this polynomial is related to the measure of each volumes via this explicit thing here so it's zero order coefficient is simply just modulo these quantities here the measure of each volume so in other words you know you can construct this function via geometric recursion and then when you integrate it you get this and so since you always have recursion when you integrate so one of the tools that geometric recursion can be used for is to show that something is satisfying topological recursion because we have this theorem that says that if you can find some function that satisfy geometric recursion its average will satisfy topological I come to that in a second so far n is bigger than one here you're equal to but I come to that on the next slide not this one but actually so if you run through the recursion what does the recursion actually say in this case here well if you look at this guy here was a polynomial in fact here is exactly how it is a polynomial and here are the coefficients and so we can write down a recursion that just involves the coefficients so you start out by setting these to be zero you put this to be this delta function and then if one one is this guy here with this C2 of two and then you have this recursion relation here with the B's and the C's and the B's and the C's are given explicitly like this where you see even C's is going in in linear and quadratic way so to answer your question if the surface of genus G and at least some boundaries the measure of each volume is given so this this is what I just said before but it turns out actually that if you have a closed surface of genus at least two then you can actually get the miss of each with no mark points by this coefficient here so you put L equal to one and you look at one puncture and you're correct with this factor so this means that you know we can very quickly you know computed via topological recursion or if you like just using this recursion relation here we can give you all these polynomials they could be put on a computer and spit out as high G and N you want okay so so far all my collaborators have been on board with this and and so this so far I've just given you things we we really have to have results and if you now allow me to speculate a little bit then the next stuff will not so if you if you have complaints about this and if you have some I'd like to hear it but don't blame my co-authors for it please okay so I I just want to try to make some speculations about the closed string field theory so in closed string field theory you have a vertex Hilbert space which comes from conformal field theory right and you have an inner product on it and the whole key about this is that this theory provides you with brackets which is indexed by G and N and some positive real epsilon which are you know multi linear brackets from V to the end to V and all of these brackets should satisfy the quantum master equation now I'm not going to write down what a quantum master equation is but just tell you that I can you know I can turn of course I can use the inner product up here to turn these brackets into brackets like this if I like so now they're just complex valued and I just the equivalent to these brackets here via this inner product like that I just dualized the first variable and then the whole sort of expectation or the thing that you or the definition of whatever you want is that these multi linear pairings are determined by integrating certain top forms over some subsets of modular space MgN not bar here but some subset inside the open part of the modular space and this subsets here together with the natural properties that these forms satisfy when they come from conformal field theory is you know the quantum master equations for these guys here is equivalent to that these subsets of modular space is satisfies this here under of course natural identification of the forms under such gluing so you're supposed to find some subsets somehow such that the boundary of these subsets for Gm is obtained this recursive way here which reminds us a lot about what we've just seen and certainly a lot about topological recursion because these are exactly the terms in topological recursion okay so what I claim is that we can actually generate a function that had that is a characteristic function on the modular space for a set that satisfies this analog of the quantum master equation here and so it's very very simple you just take the following function from r plus to r it's a measurable function it is t in the interval from 0 to epsilon and it is 0 otherwise and I require that epsilon has to be less than ox into one now I take initial data which is just a abcd coupled to these functions here and it is the should have been Mirza County McShane coupled to these f's here okay and now if you for each point in teichmann space define n epsilon sigma p to be the number of simple closed theodistics of lengths shorter than epsilon then theorem and so that one that part is okay that's easy to check actually that if you compute the amplitudes in geometric recursion for this function here coupled to the Mirza County McShane initial data you will exactly get one plus t to this power here and so in other words if I take t to be minus one the result will exactly be the indicator function of some subset and what is this subset well this subset is simply just the systle set just the subset of modular space for which the you know the all the simple closed interior theodistics have length at least epsilon and so if you analyze this function here and look at its derivative in a distributional sense you will exactly see that it's bound these sets here if you take this set here and hit it with require that all the boundaries also have length epsilon then these sets here exactly satisfy the quantum master equation so that of course tells you sorry that tells you that if you want to do this integral here you just take and multiply this guy onto these forms and integrate over the whole space and so therefore I think it's very likely that one should further be able to build the whole form here via geometric recursion because these forms here satisfy what you expect from you know conform field theory they satisfy factorizations and that is sort of how we prove that you know the function times the vapidism-sumplictic form because the vapidism-sumplictic form satisfy natural factorization rules you know we know that that we can do the integrations and we get topological recursion so I really expect that these brackets here will satisfy topological recursion proved via geometric recursion construction of this okay that would mean that we can compute them right so then we should be able to compute these and we have a recursion relation on these and so on so people who know something about string field theories would please tell me about what's what examples needs to be done here because I think this could be doable okay another example which I think one should really look at is a Kalabiyaw many folds so you should take M to be the modular space of complex structures on the Kalabiyaw you take the standard line bundle that this guy has gotten by taking the three zero part of the homology and then you're supposed to have these FGs which are the genus G topological string amplitudes which is supposed to satisfy the BCOV holomorphic anomaly equations like this you should take the mirror and then there should be holomorphic sections of these line bundles which are the genus G Gromov-Witton potentials of the mirror and then the mirror symmetry states somehow that you know FGs can be obtained from some kind of large complex structure limit of the curly FGs and so this has been understood in genus zero but certainly not in higher genus as far as I understand in general yeah yeah yes and so what I would like to understand is can we expect that these FGNs these FGs here can be given by geometric recursion so I'm trying to work on this but for now it's only just an expectation but of course then we can go on and say well Witton really had this very nice intriguing interpretation of the BCOV equations right what he did was he considered this set here which we also saw in Bertrand's talk a lot and the thing is that then this set here satisfies in this case here the heat equation and so of course if you just continue following this analog the heat equation is really a special case of parallel transport equation with respect to the Hitchin connection where you have states in some spaces of holomorphic sections parameterized by some space T which is parameterizing complex structure some pre-quantizable symplectic manifold compact with a line model over it and so I would say that one should expect that this parallel transport problem here can be solved in this form here again where these FGs here should maybe be constructable via geometric recursion maybe one should first go and restrict to the case where this M here is the modular space of flat SUM connections on the surface so I don't know but I would like to discuss this with people if they're interested I can say the following or let me be more precise if you like oops sorry take X to be a compact oriental three manifold boundary sigma consider the modular space of flat SUM connections on sigma take the transimons line bundle take the complement of the zero section of the dual line bundle then you can embed all the sections of L to the K in one and the same space namely functions on that total space that transforms in the case guy for the case power therefore in that space it actually makes sense to sort of consider the total quantum transimons state series like this which is now holomorphic function on the total space of this line bundle here of course all of it parameterized by Teichmuller space and so the conjecture would say that the total state series are resurgent and it has its singularities contained in the set of exponentiated transimons values and the zero section inside the pullback of this line bundle to the SLNC modulized space via the cotangent vibration using non-abelian hodge and then you should look at modulized space of flat SLN connections on X and compute their transimons values and so this is should be where this guy has its singularities so I think tomorrow we will see maybe an analog of this in the case where the surface is empty namely for compact manifolds that Starwars might talk about but I don't know anyway all I can say is that actually this conjecture here really works if you look at the complement of Toros knots in S3 and that's actually related to some work that I have done very recently with my PhD student William Miskard and there we established for cypher fiber manifolds that the Otsuki series is resurgent and that the singularity set is a subset of the classical SL2C transimons values and that part is on the archive this part here so this this guy came out some four months ago William or one yeah so yeah William is here you can talk to him about it Starwars just told me I have to finish very quickly higher teichmuller theory could probably also be used if you want to get at the Selberg trace formula via this I don't think you can cut along pair of hands for the same reasons what you should do is you should develop a theory where you extract triangles instead of pair of hands then you will have corners on the boundaries and this will be kind of an open geometric recursion we're working on developing this and I would think that we can prove after we've done that that the Selberg trace fits in that scheme and we can see that that would satisfy open geometric recursion therefore get same kinds of results on averages and such things thank you very much the problem is constructing a steady structure I would like to hear more details about that in your joint paper that's out already I mean yes okay but you say it you don't think it satisfies yeah questions
This is joint work, most of it, from the start quite different from topological recursion and and the morphisms are just isotopic classes of group invariant part of these vector spaces. And example of continuous functions on Teichmann surfaces so it will end with Euler characteristic abstractly length functions which are defined on just assume abstractly that there is this functorial. In particular this state that we get with the target theory being continuous functions geometric recursion will be integrable over the a mapping class group invariant measure and it's to be the number of simple closed theodistics of via geometric recursion construction of this okay