4/4 - Scattering diagram, comparison with Gross-Hacking-Keel-Kontsevich, applications to cluster algebras, applications to moduli spaces of Calabi-Yau pairs. - We show that the naive counts of rational curves in an affine log Calabi-Yau variety U, containing an open algebraic torus, determine in a surprisingly simple way, a family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a multilinear form. This is directly inspired by a very similar conjecture of Gross-Hacking-Keel in mirror symmetry, known as the Frobenius structure conjecture. Although the statement involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of U. We establish various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when U is a Fock-Goncharov skew-symmetric X-cluster variety, we prove that our algebra generalizes, and in particular gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich. The comparison is proved via a canonical scattering diagram defined by counting infinitesimal non-archimedean analytic cylinders, without using the Kontsevich-Soibelman algorithm. Several combinatorial conjectures of GHKK follow readily from the geometric description. This is joint work with S. Keel; the reference is arXiv:1908.09861. If time permits, I will mention another application of our theory to the study of the moduli space of polarized Calabi-Yau pairs, in a work in progress with P. Hacking and S. Keel. Here is a plan for each session of the mini-course: 1) Motivation and ideas from mirror symmetry, main results. 2) Skeletal curves: a key notion in the theory. 3) Naive counts, tail conditions and deformation invariance. 4) Scattering diagram, comparison with Gross-Hacking-Keel-Kontsevich, applications to cluster algebras, applications to moduli spaces of Calabi-Yau pairs.
 Thank you for coming to the last session of this mini course. Let me first share with you the files in the chat and also... Yes, so let's start. Here is where we were on Monday. The idea of the gluing formula is as follows. Roughly the gluing formula states that we can glue two open curves along two opposite boundary components and form a bigger concatenated curve like this. And the count of the concatenated curve should be the product of the counts of the two initial curves by this formula. So this is more convincing evidence that our counts really reflect open curve counting. It is also an essential ingredient in the proof of the associativity of the mirror algebra. Now let's establish this gluing formula in three steps. Step one, given two spans S1 and S2 in the essential skeleton of U, assume we have an infinite one-valent vertex W1 and an internal infinite one-valent vertex, meaning that it's mapped to the essential skeleton instead of to the boundary. So assume we have such a vertex W1 in S1 and respectively we have an internal infinite one-valent vertex W2 in S2 such that the two vertices map to the same point in the essential skeleton SKU. Next we consider a span delta with three infinite one-valent vertices W1, W2, W and mapping constantly to this point where the previous W1 and W2 go. So we consider this span which maps constantly to the same point and this span has just one three-valent vertex and three infinite legs. So we can glue this constant span delta to S1 and S2 along W1 and W2 and form a new span S like this in the essential skeleton of U. Here the infinite one-valent vertices W1 and W2 they become nodes in the new span after we glue. So in the new span this is an infinite edge containing a node and this is another infinite edge containing a node. Now we have the following M. For any curve class gamma we have the following equality that is the count associated to the span S and the curve class gamma by evaluating at this internal marked point W is equal to the sum over all the compositions of gamma into gamma 1 plus gamma 2 of the count associated to the span S1 curve class gamma 1 times the count associated to the span S2 curve class gamma 2. So for S1 we evaluate at the point W1 and for S2 we evaluate at the point W2. So this is the formula we have for the count of the glued span since this part is constant it doesn't contribute. The proof is not difficult by passing to a big enough base field extension this follows from a set theoretical decomposition of the set of skeletal curves associated to S of the left hand side to products of sets of skeletal curves associated to S1 and S2 respectively. So after big enough base field extension it reduces just to a set theoretical equality one can just check by hand. So this is the first step of gluing we glue two spines with this auxiliary spine at these infinite vertices. Now let's consider the second step of gluing. So in the second step we are given two spines S1 and S2 in the essential skeleton of U both transverse to walls and assume we have a point P1 in gamma 1 the domain of S1 and P2 in gamma 2 the domain of S2 both in the interior of some edge and we assume that they map to the same point in the essential skeleton and also they do not meet wall. Since they map to the same point we can glue so we can glue S1 and S2 along the points P1 and P2 and obtain a new transverse spine S in the essential skeleton of U like this we just glue these two together at the point P1 and P2. Then we have a similar lemma saying that for any curve class gamma we have the following equality which says that the count associated to the glued spine S and any curve class gamma is equal to the sum over all the compositions of gamma into gamma 1 plus gamma 2 of the count associated to the first spine S1 and the curve class gamma 1 times the count associated to the second spine S2 and the curve class gamma 2. Here is the proof for this for this equality so for the proof we add an infinite leg W to S at P because if you recall that in our definition of counts we always need some internal mark point in order to evaluate and we proved using skeletal curves we proved a symmetry theorem saying that the place where we evaluate eventually does not matter but for the just the first to construct the count we always need some internal mark point so let's add it let's add the infinite leg W to S at P at this the point we glue P and then we count this spine by evaluating at this point W and after adding this infinite leg now we can deform by stretching this point P and make appear two small edges E1 and E2 so this is a small deformation of this by stretching the point P and that now W is attached the leg W is attached to the middle of this purple of the two new purple edges and if we further stretch if we further stretch the two vertical edges E1 and E2 to make them of infinite length and contains contain a node so we make each edge infinite length to contain a node then we arrive at the gluing situation of step one and note that in this stretching process all spines are transverse so their counts do not change by deformation invariance and now we just conclude by the lemma we proved in step one we show and we conclude our proof for this equality so this is the second step to establish the gluing formula it's it uses a deformation invariance for transverse spines as well as the previous lemma that we established for the counts of such a glued span with this auxiliary with this auxiliary span delta in the middle so finally in step three for the gluing formula we are given two spines S1 S2 in the essential skeleton of you both transverse to wall like this S1 S2 and assume we have finite one valent vertices V1 in the domain gamma 1 of S1 and the V2 in the domain gamma 2 of S2 such that they map to the same point in the essential skeleton of you and we assume that we have opposite derivatives meaning that the derivative at V1 of H1 plus the derivative at V2 of H2 is zero since they have opposite derivatives and they map to the same point in the essential skeleton of you we can glue S1 and S2 along V1 and V2 and obtain a new transverse span S like this in the essential skeleton and here is the final theorem the final gluing formula for any curve class gamma we have the following equality which says that the count associated to this glued span S and any curve class gamma is equal to the sum of all the over all the compositions of gamma into gamma 1 plus gamma 2 of the count associated to the left part S1 with curve class gamma 1 times the count associated to S2 and the curve class gamma 2 so this is the final gluing formula and it is a generalization of the two dimensional case in my previous paper but here I'm presenting a more conceptual proof via deformation invariance so the proof the idea of the proof is the following so we want to prove that when we glue these two together we have this formula for the count and let's make a small extension of S1 this is our S1 we make a small extension of our S1 at V1 to S1 hat by linearity so we just extend linearly at this vertex V1 extend a little bit this purple part is our extension and similarly we make a small extension of S2 at V2 by linearity to S2 hat and by deformation invariance for this transverse truncated spice the count remain the same the counts do not change when we make these small extensions as long as we do not meet walls and now after making the two small extensions observe that if we glue S1 hat and S2 hat together by identifying V1 and V2 so let's glue S1 hat and S2 hat together at V1 and V2 we see that once we do that this gluing is actually just equal to the glued span S to the gluing of S with some small straight span L so this gluing S1 hat S2 hat at V1 and V2 it's just it's equal to the gluing of S this red S together with this purple purple edge L at the point V and both sides of this formula are gluing of two spans so we can so both sides are gluing of other such gluing situation of step two so we can apply step two to both sides of the above equality and we obtain this which says that the sum over all the compositions of gamma into gamma 1 and gamma 2 of the count associated to S1 hat and the gamma 1 times the count associated to this extended span this small extension S2 hat and gamma 2 is equal to the sum over all the compositions of gamma into beta 1 plus beta 2 of the count associated to S this red S and the curve class beta 1 times the count associated to the L this purple small straight span L and curve class beta 2 so now we have to let's compute explicitly the contribution of this part so we can explicitly compute that the count associated to L and the beta 2 is equal to 1 if beta 2 is 0 and it's equal to 0 otherwise and if we substitute if we substitute this explicit computation into the equality above we obtain the gluing formula in R0 sorry do you understand you have this purple interval L maps to a point is essentially the only contribution you have yeah L doesn't map to a point L map L maps to a small interval a small interval I see yeah yeah so it's really yeah it's not clear what does correspond to in simpleclic topology because you don't consider p1s but kind of like annular yeah it's yeah it's a very small annular map into a place without walls I see any walls I see yeah so that place is like just a torus so we have it's like we count annulus just in an algebraic torus yes so the count is really the simplest yeah either one if a curve class is zero or zero if it has if we put some non trivial curve class because it it's just a maps to a place without walls so there is nothing interesting happening yeah so via these three steps we obtain this gluing formula zero and let me remark that similar idea can be applied to show that our accounts are independent on the choice of the torus when we impose the toric tail condition assume we have two torus embeddings tm in you and the tm prime in you just two different embeddings leading to two different toric tail conditions t and t prime so recall that in the definition of our naive accounts of skeletal curves in order to obtain a finite dimension or modular space we need to impose some extra condition and the some extra regularity condition on the boundary via analytic continuation and that the condition called the toric tail condition was formulated according to the choice of some torus torus embedding now we want to show that it's independent of the torus embedding using the same idea in the proof of the gluing formula so assume we have two torus embeddings tm in you and the tm prime in you leading to two different the toric tail conditions quality and quality prime and now we consider a span s like this in the essential skeleton of you with a finite one valent vertex v so here the span s is the whole graph including the purple part this is our span s it has a finite one valent vertex v so we want to show that the count of this span s does not depend on which tail condition we impose at this end v for this let's pick w some point w very close to v and let l denote the restriction of the span s to this small interval w v so this purple interval is just a restriction of s to a small neighborhood of v and then we pick any point x in the middle of w and v and we consider the gluing of s with l so it's similar to the gluing we considered a moment ago here when we consider gluing of s and l we get like this part gets doubled it's like they're double now by step two above we obtain the following equality so here you see that we are gluing we are gluing two spans at some interior point of edges which is the situation of step two and so we can apply the formula in step two and now if we apply the formula in step two we obtain the following so the left hand side yeah so when we apply we think like this for the the left hand side is the sum over all the compositions of gamma into beta plus delta of the count associated to s using the tail condition the first the tail condition t everywhere times the count associated to l using the tail condition t prime at v and t at all at the other end of l so this left hand side so we can think of l as for the left hand side we put the toric tail condition t everywhere on s but for l we put the toric tail condition t at w but tail condition t prime at v and then we apply the step two above to this gluing now for the right hand side we just think that for the gluing we switch we switch this part of l with this part of s which means that it's the sum over all the compositions of gamma into beta plus delta of the count associated to our span s where we use the tail condition t prime at v and t in all other places now at times the count associated to this small interval l where we apply tail condition t at both ends so it's really so the left hand side as I said the difference between the left hand side and the right hand side it's how we think of the gluing it's like we are switching that half of l with that last piece of s so one piece has the toric tail condition t attached and the other piece has toric tail condition t prime attached when we switch them in this gluing we obtain and when we apply step two we obtain this we obtain this formula then this equality so now it remains to compute explicitly the computations of the small this l piece so we can compute as before that the count associated to l using the tail condition t and the curve class delta is equal to one if delta is zero and if it's equal to zero for all other delta and the same holds when we count the small interval l using toric using tail condition t prime at v and t at w the same equality the same count works also for the other one and we now we substitute we substitute this explicit computation into the equality above we obtain the following theorem for tail condition with varying torus so we have proved by substituting this explicit computation into the equality we have proved that the count associated to our spine s using tail condition t everywhere and and for some fixed class gamma is equal to the count associated to the spine s equal to the count associated to the spine s using tail condition t prime at v and tail condition t everywhere else these two counts are equal so here we are just switching tail condition at one vertex then of course one can switch at all other vertices and we can even apply different toric tail conditions at all finite vertices so this in other words the theorem shows that the count of skeletal curves is independent is independent is independent on the choice of the torus tm inside u so just a small remark concerning the computation for the count associated to this small interval where we put different toric tail conditions at both ends so the explicit computation for this count using two different tail conditions is a bit more subtle than for the count where we put the same tail conditions we need to use a result concerning the gluing of non-archimede polyannual in my previous paper if we put two different tail conditions because if we put the same tail condition then the count is easy it's like something yes sir you know maybe you mean that on one part of interval every put condition t prime another t but your notation your notation don't say this it's kind of like t prime everywhere yeah yeah but actually it says that it's one end is t the other end is t prime i see so i hope that it's understandable okay otherwise the notation is a bit complicated yeah so i'm saying that if at both ends we put the same tail condition then the count is easy to do because yeah yeah as maxi remarked we just map to a torus so then it's just counting something in the toric variety but if we put different tails at the two ends then the counting is a bit more complicated because it maps to some gluing of gluing of two different torus but one can show that actually one can decompose the the automorphism of the some annulus into two parts where one part can be extend can be extended to a automorphism of some disk times poly annulus and another part can be extended to automorphism of another opposite disk times the annulus so after applying this automorphism we see that we are we we are back to the previous situation just counting something in p1 times some poly annulus the computation of automorphisms of this automorphisms of the some affinoid algebra yes yeah so this is the idea for the proof of the gluing formula and also for the proof of changing tail condition and in fact this proof of changing tail condition also works if we eventually use more general tail conditions without using any embedded torus so it's easier to carry out this proof than one can imagine using like deformation variance for it feels a bit more complicated so now let's turn to the next section structure constants and associativity of mirror algebra we want to apply our gluing formula and all the other techniques we have developed so far to study structure constants and the associativity of the mirror algebra so let us recall the setting from the first lecture where we have log kala b y of variety u containing some torus tm and we have some s and c compactification of u in y we have monoid ring r over this effective curve classes assembling all possible curve classes together and the mirror algebra a it is it has basis the integer points in the essential skeleton as an arm module so this was our setup in the first lecture and again recall that given some integer points in the skeleton we write the product in the mirror algebra a as the this product of the theta functions theta p1 to theta pn in the mirror algebra a as this sum so first we sum over all integer points in the essential skeleton of the basis vector theta q and then we sum over all curve classes over all effective curve classes of the basis element z to the gamma it's just a notation for the basis elements and we denote the coefficient by chi p1 to pn q gamma and this chi is called the structure constants of our mirror algebra a and let's also recall how it was defined in in the first lecture the structure constant chi was defined as follows we had so first we figure out what was the class delta of the editory tail and we denote the total class beta to be gamma plus delta beta is supposed to be the class of the the closed p1 after the total tail extension we also had the z equal to the opposite of q in the essential skeleton and we had topo pz a putting p p1 to pn and z together and we use this topo pz to specify intersection numbers with the boundary then we considered the modular space this hpz beta with marked points labeled as p1 to pn zs so this is the modular space of maps of maps from p1 with marked points p1 to pn zs and we say that we specify the intersection of the p1 so it's the modular space of maps of p1 into y and we specify the intersection numbers at these marked points with the boundary d using the topo pz and we had a natural map phi from this modular space to here taking domain taking domain and evaluation at the last marked point s and we also had a special point qtodah in the identification of the target which was given as a pair of mu and q mu we give mu we just specify what was mu in the first factor corresponds to some divisory evaluation and q is just the given q so the given q is the integer point in the essential skeleton in particular it's a point in the identification of u so we had this special point qtodah and then finally we had a subspace f in the fiber of the map phi over qtodah which is a finite analytic space and if we take its length the length was by definition the structure constant kind of a in the first lecture first I give a heuristic picture of what we count for the structure constant about counting disks with some conditions on the derivative of the disk at the boundary and and after that I give a precise construction of the structure constant in using algebraic using noic median geometry so this was just a recall of what we did and the definition is quite straightforward but let's remark the following due to the choice of this specific point qtodah inside the target so we used this qtodah to take fiber and then take subspace due to the choice of this specific point the curves in f are responsible for structure constants although highly generic in the algebraic sense are in fact very special in other words non-transverse from the tropical viewpoint because this special point it's a generic point in the algebraic world but it's a very non-generic point in the tropical world so this results very generic curves in the algebraic sense but very special curves from the tropical in the tropical picture and this was convenient for giving a quick definition of structure constants but it's impractical for proving any properties about them for example associativity finiteness all these properties they are out of reach from this quick definition we must deform the curves in f into more transverse positions by perturbing this special point qtodah because when we want to prove properties for example if we want to apply the gluing formula or deformation invariance we usually need the assumption that the spines are transverse but if we do the count using this specific point qtodah we will not get transverse spines so we must perturb these curves by varying the point qtodah do we do it like this for a position we label the marked points of metric trees in this modular space so this is a modular space of tropical curves rational tropical curves with n plus one n plus two legs here abstract tropical curves are the same as metric trees and we label the n plus two points marked points as p1 to pn zs and let vm in the modular space be the subset consisting of metric trees whose z leg and s leg are incident to a single three valent vertex here is the picture we have such a metric tree it has a lot of legs by leg we mean infinite one valent vertex or yeah so or more precisely we mean these edges containing infinite one valent vertices so here we consider the subset where the z leg and the s leg they are incident to a single three valent vertex so the z leg and the s leg they meet first and before the tree branches to other legs yeah and we observe that this vm is is a neighborhood of this special choice of the modulus mu and next next we want to figure out a neighborhood of this special point q so let's consider a polyhedral subdivision sigma of the essential skeleton given by the set of walls in the essential skeleton here we can assume the set of walls to be finite polyhedral by bounding the degree of twigs by the fixed curve class beta so in general uh wall the set of walls they are infinitely many walls and it can be dense in the essential skeleton but if we bound some degree we get a finite set of walls so we consider the polyhedral subdivision induced by the set of walls and we let vq be the open star of the point q in sigma in other words the union of open cells in sigma whose closure contains q so we have q and we have a polyhedral decomposition we just take a cone around around the point q q might be q might lie inside a wall but it doesn't matter we take the star of q in sigma then q becomes an interior point in this open star and we said that by construction q tilde which was given as a pair mu q mu is the interior vm is a neighborhood of mu vq is enabled of q so q tilde so vm times vq is a neighborhood of q tilde inside the product of the tropical modular space and the essential skeleton and we already remarked when we talk about essential skeletons that using Tamkin's matrization theory one can show that product of skeleton is homomorphic to skeleton of product and this lies in the identification of the product yeah so using this construction we figure we figure out two natural neighborhood of sorry we figure out a natural neighborhood of the first factor of q tilde and a natural neighborhood of the second factor of q tilde so we figure out a neighborhood of q tilde i recall that our goal is to perturb the point q tilde so we will be perturbing q tilde inside this neighborhood vm times vq and now let this be the pre-image of vm times vq by the map phi analytic from this modular space by taking domain and evaluation at the last mark point s so before we took pre-image of a single point q tilde now we take pre-image of this neighborhood of q tilde and now we take curly f to be the subset in the pre-image satisfying the toric tail condition then the preposition says that phi analytic is finite at all on the neighborhood of this subset and whose degree gives the structure constant so this is how we perturb the special point q tilde into general position by allowing q tilde to vary inside this neighborhood vm times vq and we prove that if we allow q tilde to vary and when we impose toric tail condition then this map phi is still is good it's finite at all on the neighborhood of this f so the degree is well defined and it gives the structure constant since it's finite at all so we get a well defined degree and the the degree at the fiber over the point q tilde it was our quick definition of the structure constant so here the structure constant is reinterpreted as some degree of finite at our map and for its proof we use the toric tail preposition for almost the transverse spines because here it feels like deformation invariance that we are moving we are moving the spines for the structure constants inside this conical neighborhood but they do not state it's not always a transverse sometimes it it becomes non-transverse especially at the point of q tilde for example but we had developed this reposition in the last lecture not just for transverse spines but is that it is specially adapted to the situation here where we can go across the walls so after this perturbation into generic position now we can prove the following theorem the multiplication rule given by the structure constants chi here is commutative and associative here is the the sketch of proof commutativity is obvious because the definition of the structure constant chi is symmetric with respect to the p i's and associativity means that the product theta p1 theta p2 to theta pn does not change if we add arbitrary parentheses so let us now sketch the proof of the following equality where the left hand side means we first take product of theta p1 and the theta p2 and then we take product with theta p3 while the right hand side means we take the product theta p1 theta p2 theta p3 together using the multiplication rule so we want to prove this equality and we just rewrite the products using the multiplication rule and we substitute the multiplication rule into the equality and we see that the equality becomes equivalent to the following equality for every integer point q in the essential skeleton and every curve class gamma now observe that the right hand side of the equality star is given by accounts of skeletal curves associated to the spines of this shape where we have three infinite lags with derivatives p1 p2 p3 respectively and also we have one finite lag with the inward derivative q so three infinite lags with outward derivative p1 p2 p3 and one finite lag with inward derivative q and by the above proposition about perturbing q tilde we can deform the modulus of the domain here by stretching this point so we deform this point into a small path l and then we further stretch this path l very very long like this so if we stretch the path l very long we see that the point u near the top of the path l will map sufficiently close to the ray zero r inside the essential skeleton of u and we need it to be sufficiently close to the ray because we want it to lie inside the cone vr in the essential skeleton of u in as in the above proposition for the structure constant chi p1 p2 r eta recall that in our quick definition of structure constants we say that the marked point just go directly to the point q tilde and that was two non-transverse two rigid now we allow q tilde to move a little bit around but we still we always need it to be sufficiently close to the ray oq so it it should not move outside across some walls around this ray otherwise it doesn't give the correct structure constant so here we stretch this path l very long so that finally some point u near the top of the path will be sufficiently close to the ray or then then that will be good enough for defining the structure constant chi p1 p2 r eta so we can cut at the cross u apply the gluing formula and obtain the left hand side of the equality star so similarly given two spines responsible for the product in the left hand of the equality star we can glue them to form a highly stretched spine like this that is responsible for the right hand side and this completes the proof of associativity so before the break let me quickly sketch another important property of our structure constants which is the convexity property so here is the theorem of the convexity property let f be a Cartier divisor on y and consider consider an analytic disc in general position responsible for the structure constant for the structure constant chi p1 to pn q gamma then the following hold first since f is a Cartier divisor we can take its tropicalization and obtain a real valued function f chop on the essential skeleton then first we have that the sum of f chop at overall pi minus f chop at q is equal to the intersection number between f and gamma minus the degree of f and liitified restricted to the punctured disc meaning the disc minus all the marked points and the second if f is an f and the minus f restricted to you is effective then we have f chop of q is less than or equal to the sum of f chop at every pi and furthermore assume f is ample and the minus f restricted to you is effective then the above equality inequality is an equality if and only if f maps the punctured disc into the torus and the proof uses some detailed computation using semi-stable models of curves the convexity theorem implies the following finiteness result finiteness result one given p1 to pn in the essential skeleton of you some integer points in the essential skeleton of you then there are at most finitely many pairs q gamma where q is an integer point in the essential skeleton of you and the gamma is a curve class such that the structure constant is nonzero let me give a quick proof how the finiteness follows from the convexity so since u is a fine we can find the regular functions x1 to xl on you such that the set of points in the essential skeleton where the norm of xi at b is bounded by some real number c for all i this set is a is bounded for any real number c now if the structure constant is nonzero so by the convexity statement two we apply the convexity statement to the car t a divisor given by these regular functions we obtain this equality the norm of xi of the function xi at q is less than or equal to a sum of is less than or equal to sum over all j of x the norm of xi at pj and this shows that given p1 to pn there are at most finitely many q such that the structure constant is nonzero for some gamma so this bounds q and the next let's bound the gamma we want to bound both q and the gamma and the assumption that u is a fine implies that there is an ample divisor f on y such that minus f restricted to u is effective and now we apply the convexity theorem one the statement one this statement and we obtain this the following equal the following equality and inequality which says that the intersection number between f and the gamma is equal to the sum of f drop at every pi minus f drop at q plus the degree of the analytic f restricted to the punctured disk and since minus f restricted to u is effective and this degree is non-positive so we see that this is less than equal to this and this is fixed the right hand side this bounds gamma by the amponeness of f so this is how we deduce the finiteness result from the convexity property and the finiteness result is important because it implies that the two sums in the multiplication rule here they are finite sums so the multiplication rule gives an algebra structure on the free r module a instead of just some formal algebra structure and in fact we have the following stronger finiteness result too which says that the mirror algebra is a finitely generated r algebra and for its proof we need to result to the equivalent boundary torus action on the mirror algebra so here due to time constraints I will omit this boundary torus action and finite generation result in this lecture and after the break I will explain the application towards cluster algebra and also I will explain the wall crossing how to get scattering diagrams using these counts of analytic curves so let's make five minutes break okay thank you so here is the plan for the last part of this lecture first I'll explain how to construct a scattering diagram via infinitesimal analytic cylinders and the second I'll prove the property of theta function consistency for the scattering diagram and third we will set all curve classes to zero so that we no longer care about the compactification and the fourth I'll explain the class we will apply the above to the case of cluster algebra where I need to introduce two new notions C twigs and the C walls especially for the cluster case and finally I'll explain the comparison with the work of gross hacking kio condo savage on the for class with the work of gross hacking kio condo savage for cluster algebras so let's start with first scattering diagram via infinitesimal analytic cylinders both in the yes both in the original suggestions by condos savage soba man and in the gross seabird program the construction of a mirror variety relies on the combinatorial algorithmic construction of scattering diagram also known as wall crossing structure our construction our construction of the mirror algebra by counting non-archimedean analytic disks as in the previous lectures completely bypasses any use of scattering diagram nevertheless our geometric approach also allows us to give a direct construction of the scattering diagram by counting infinitesimal analytic cylinders without the step-by-step condos savage soba man algorithm and this has three implications first it gives a geometric interpretation of the combinatorial scattering diagram second conversely we obtain a combinatorial way for computing the non-archimedean curve counts and third it paves the way for the comparison with the work of gross hacking kio condo savage on cluster algebras let me also remark that there are recent works of argus gross gross seabird gross seabird give another geometric interpretation of scattering diagram based on the theory of punctured log curves developed developed by abramovich chain gross and seabird now let us sketch our construction of the scattering diagram via infinitesimal analytic cylinders recall we have our log halabi au containing some torus tm and is contained in some s and c compactification y and we denote by n the dual of m so definition given a hyperplane n-purple in mr m tensor with r and any generic point x in the hyperplane m-purple generic means that it's not contained in any other n-purple it's only contained in this one hyperplane and say we are given two vectors v and w in m minus n-purp and the curve class alpha let v x v w be the infinitesimal spine here bending at x with incoming direction w and outgoing direction v and this gives rise to the associated count of analytic curves n v alpha so we consider this infinitesimal spine bending once at a generic point x in the hyperplane m-purple with specified the incoming and outgoing direction and we consider the associated count of analytical curves then using all these counts for any x in some hyper for any generic point x in some hyperplane and perp we define the following wall crossing transformation and psi x n acts on the basis vector z to the v as follows if we if so for all v in m which pairs with n positively we define the value of psi x n at z to the v to be the sum over all possible vector of all vectors of every vector w in m which pairs with n positively and over all curve class of the basis vector z alpha z w with coefficient the count we just defined so in other words we just sum over all possible incoming direction w and all possible curve class and if for all v in m which pairs with n trivially which lies in the hyperplane we just defined it to be z to the v it doesn't the wall crossing transformation does nothing for z v and unlike in the multiplication rule this sum does not is not a finite sum but this converges in the natural eddyc topology which we describe here since this cone of curves may not be polyhedral we fix a strictly convex toric monoid q containing this effective curve classes and let our hat be the completion of the monoid ring over q direct sum m with respect to the maximum monomial ideal i in other words the ideal generated by monomials z qz m with q nonzero and arbitrary m so we use this completion to express the convergence lemma convergence lemma the formal sum in the wall crossing transformation lies in our hat so the reason is that when we give bound on curve classes it implies bound on the combinatorial types of the two-weeks of all analytical curves contributing to these counts and this in turn gives a bound on the incoming directions w so now by linearity over this monoid ring we can extend the wall crossing transformation transformation to a map from the monoid ring over all curve classes plus direct sum with the subset of the lattice m which pairs with m nonactively and the map goes to our hat we just here we defined what it does on the basis vectors and we extended by linearity over the monoid ring theorem wall crossing homomorphism theorem the map psi xn is a ring homomorphism so it means that if we pick arbitrary m1 m2 in m that pairs with n nonactively we want to show that we have a psi applied to z to the power n1 plus m2 is equal to psi of z to the m1 times psi to the z of m2 it's what means for the map to be a ring homomorphism so let's try to prove this we let's fix any vector e in m and any curve class alpha and let's prove the equality of the coefficients of z alpha ze inside the formula above note that it's obvious the equality is obvious when both m1 and m2 lies in n-perf because if they lie in they lie in n-perf then the wall crossing transformation does nothing on them so we may assume that one of them pairs with n positively so assume that the pairing between n and m1 is positive and the pairing between n and m2 is non-strictly positive and we consider the count of analytic pair of pants associated to this spines with three just various infinitesimal spine with three legs with three ends near x with directions m1 m2 minus e at the three ends so claim suppose x is contained in the wall sigma then the count of analytic pair of pants is independent of which side of sigma the three valent vertex of this our spine maps to here the left picture shows this spine mapping to the essential skeleton and this blue blue line is a wall and the left picture shows the situation where the three valent vertex maps to the left the three valent vertex maps to the left and the right picture shows the situation where the three valent vertex maps to the right so the claim says that the count is independent of which side the three valent vertex go let's further observe that in the left picture when we specify the directions m1 m2 and e this the shape of the spine is unique which we denote by sl because there is only one possible band and that band is determined by the three directions at the ends however in the right picture the shape of the spine is not unique because we have two bands and this band when we deform from left to right this band is decomposed into two bands and all possible ways of decomposition are allowed so for the right hand picture we are actually in summing over many we are summing over many different shapes of spines which we denote by sri sorry Tony yeah the source stability is even m2 is parallel to the wall which is yes yes this is the trickiest this is the trickiest possibility but i allow m2 to be parallel to the wall here what is your doubt where is your doubt no doubt no no it's allowed yeah yeah and it's exactly the possibility where m2 is parallel to the wall that possibility implies that like the that the preservation of volume elements we see so yeah so we keep in mind that we also have the possibility m2 parallel to the wall yeah so as Maxim remarked that here in the proof we use the proposition for toric for tail conditions in families in the last lecture and the trickiest case is when one of m1 or m2 lies in m-purp one of them is parallel to the wall where we need to use the deformation invariance for almost the transverse suspense so now let's go back to the proof of the theorem this is proof for the claim about deformation invariance when we move from left to right and now let's back to the proof of the wall crossing homomorphism theorem note that for any span l disjoint from walls the count nl gamma is 1 if gamma is 0 and is 0 otherwise because if it's disjoint from walls we are essentially in the toric situation and everything can be computed explicitly keeping this in mind now we consider first the left picture if we cut the spine sl at the cross by the grouin formula then by the grouin formula the count associated to s alpha sl this red span sl and any curve class alpha gives the coefficient of z alpha ze in this wall crossing transformation because when we cut at this cross the left part is disjoint from wall so it doesn't contribute while the right part has a band and the right part is exactly the infinitesimal cylinders that we use in the definition of the wall crossing formula so therefore when we count analytic curves associated to sl we get the coefficient in this wall crossing transformation and the next if we cut next let's consider the right picture where the three valent vertex maps to the right if we cut the spine sri at the two crosses but then by the grouin formula so we cut at these two places now the spine is broken into three pieces note that the right piece is disjoint from any walls so the right piece doesn't contribute but the two left pieces they both contribute and they both have a band and we just add their contributions together so by the grouin formula we see that the count associated to this the count associated to this spine sri is equal to the sum over all the compositions of curve class alpha into alpha one plus alpha two of the count associated to this upper left spine times the count associated to the lower left part and we remarked that when the three valent vertex maps to the right of the wall we have different shapes parametrized by i and next we sum over all possible shapes sri and we see that the sum of counts associated to sri gives exactly the coefficient of z alpha ze inside the product of the two wall crossing transformations so we concluded the proof of wall crossing homomorphism by the claim yeah so this shows that wall crossing transformation is a ring homomorphism and we remark that the immediate consequence of the ring homomorphism is the following given any x generic any generic x in some hyperplane n perp and any vector v in m which who's pairing with n equals one so in particular this means that n is necessarily primitive vector we just rewrite psi xn at zv to be zv times some function this is just a rewriting with some function in our head then the fact that wall crossing is a ring homomorphism implies that the function f x and v does not depend on v geometrically this means that the count of infinitesimal analytic cylinders depends only on the amount of band independent of incoming or outgoing direction so for the count of this we can change incoming or outgoing direction as long as we have the same amount of band we always have the same count in particular this implies that the wall crossing transformation preserves the standard volume form on tori and moreover we have we have we have the equality between the function fx and v and fx minus and v in other words it's independent of the orientation of wall the function is independent of orientation of wall so we can denote fx so we can denote fx and v just by fx since it's independent of n and also of v for any choice of n primitive whose prep contains containing x and any choice of v with the and the pairing between nv equals one and we call fx the wall crossing function attached to x so here is the conclusion now we can write the wall crossing transformation applied to zv simply as zv times fx to the power the pairing between n and v for any for any vector v in m that pairs with n non-actively and this is the it resembles like to the more classical wall crossing formula and using this formula we see that the wall crossing transformation psi xn extends to an automorphism of the fraction field of our hat let me remark that this may not give an automorphism of our hat since the wall crossing function need not to be invertible in our hat because of curve classes it might not start with one it might just start with a curve class and that curve class doesn't is not invertible it doesn't matter first this is just how it is if we take into account curve classes and it will become invertible when we set all curve classes to zero now definition now we have all the wall crossing transformations wall crossing functions we can define what is our scattering diagram so let d be the set of pairs xfx where x is any generic point in m-perp in the hyperplane n-perp for any non-zero n and fx is the associated wall crossing function we call this set of pairs xfx we call it the the the scattering diagram associated to u with respect to the compactification y and also with respect to the torus tm so scattering diagram depends on the torus and we remarked that by the eddy convergence of fx of the wall crossing function if we mod ik where i was if we mod out some power of i where i was the maximum monomial ideal then we obtain a finite scattering diagram dk in other words finite means that we will only have finitely many polyhedral walls in dk here in d we have infinitely many walls so it no longer makes sense to say the shape of each wall they are so small it it's better just to give the wall crossing functions attached to each generic point but once we mod out some power of the maximum monomial ideal we get finite scattering diagrams dk with finitely many polyhedral walls and we call dk the case or the approximation of d and one important property for scattering diagram is the consistency consistency property so let's try to establish consistency first let us establish a variant of consistency property called which we call theta function consistency we introduce a new definition choose any generic point x in mr and the two vectors me in m and let sp x me be the set of spines in mr with domain minus infinite infinity to zero like this such that minus infinity maps to the boundary with derivative minus m and zero maps to x with derivative minus e so we consider the set of all such red spines starting from infinity ending at x with with directions m and e at the two ends and this is related to the notion of broken line in ghkk we define the local theta function theta xm to be the formal sum over all vector e and all such red spines s and all curve class of the vector z alpha the basis vector z alpha ze and with the coefficient the count associated to the the red span s and alpha and again we have addic convergence this lives in our head and we have the following theta function consistency theorem the scattering diagram d is theta function consistent in the following sense given any k we consider the case order approximation given any k we consider the case order approximation of our scattering diagram dk and we take a polyhedral wall sigma with attached wall crossing function f sigma inside dk and we choose a vector n such that the wall is contained in the hyperplane and perp and we consider two points a and b two general points near another general point x in the wall on two sides of the wall so we pick a general point x in the wall and the two general points on two sides of the wall a and b near x such that n a pairs with n positively and the b pairs with n negatively then we have the following if we apply the wall crossing transformation per se to the local theta function at a with infinite direction m we obtain the local theta function at b with the same infinite direction and conversely if we apply the wall crossing transformation per se with respect to the opposite orientation of the wall to the local theta function theta bm then we get back the local theta function theta a m this is what we call the theta function consistency property and we use the gluing formula and the deformation in variance for the proof so our next step is to forget all curve classes we want to get closer to the classical wall crossing structure or classical scattering diagram without always thinking about curve classes in our compactification so let's set all curve classes to zero as I said our wall crossing transformation and the scattering diagram depends on the compactification u inside y via the usage of curve classes in y so this is a more refined information but we can remove this dependence by setting all curve classes to zero in order to have attic convergence if we set all curve classes to zero we need to impose a condition on the band of infinitesimal analytic cylinders without extra condition we will lose attic convergence if we forget the curve classes the assumption is the following for any for any non-zero count associated to some infinitesimal spine at x with the incoming direction outgoing directions v and w if the count is non-zero then the band w minus v lies in the strictly convex monoid p inside m we need this assumption in order to have attic convergence when when we forget curve classes so and here is the new attic convergence we consider thanks to the monoid m we consider j the maximum monoid ideal in the monoid ring over p the monoid p and let l not hat be the genetic completion of zp and finally let l hat be l not hat tensor with zm over zp in other words we are taking completion of zm in the direction of the strictly convex monoid p we allow infinite sums in the p direction and only finite sums in the other direction in all other directions so for now we will be forgetting curve classes for a position under the quotient from the monoid ring associated to q direct sum p to the monoid ring associated to p when we ignore all curve classes the wall crossing functions fx in r they map to the quotient wall crossing functions fx bar in l hat meaning that when we ignore curve classes they have attic convergence in l hat and the the wall crossing transformations psi xn which used to be an automorphism of the fraction field of r hat they become automorphisms of l hat so we no longer need to take a fraction field of l hat because when we forget the curve classes the wall crossing transformations will automatically become invertible and furthermore the local theta functions in r hat they map to the quotient theta functions in l hat so we have this j attic convergence for all these things before and moreover we get the invertibility of the wall crossing transformation so now we denote du to be the set of pairs x fx bar where x is any generic point in any hyperplane and probe and the fx bar is the associated wall crossing function ignoring all curve classes and we call this the scattering diagram associated to u with respect to the torres tm now we have the following consistency result which says that the scattering diagram du is consistent in the sense of contrived sobo man in other words for any general loop inside the mr the composition of wall crossing automorphisms after ignoring curve classes along the loop is just identity so for the proof we observe that the subring of l hat generated by all local theta functions is genetically dense and then the theorem follows from the theta function consistency so that's the general constructions of the scattering diagram using counts of infinitesimal analytic cylinders and now let us apply our constructions above to the case of cluster algebra now here is the cluster data we have lattice m with an integer valued skew symmetric form and we have s prime a basis of m and s inside s prime a subset of s prime this gives a seed for a skew symmetric cluster algebra of geometric type where s corresponds to unfrozen variables and the s prime minus s corresponds to frozen variables and the seed gives rise to curly a a fork contour of a type cluster variety which is the gluing of torri via cluster mutations and we let a up b the algebra of global functions on a called the upper cluster algebra let us assume that the skew symmetric form is unimodular we assume this for mainly for con partially for convenience this holds in the principal coefficient case because this holds in the principal coefficient case and then we will deduce more general cases from the principal coefficient case and we also assume that the spec of the upper cluster algebra which we denote by u is a smooth case so for example double brouha cells in semi-simple complex complex three groups satisfy this smoothness assumption I think it's possible to extend our work also to cover the non-smooth case without too much extra effort but I haven't checked all the details then u is a log halabi all containing a torus dm so we can apply our theory to u and obtain a mirror algebra as well as a canonical scattering diagram by counting non-archimede and elliptic curves and here is the question how shall we compare with the constructions in the paper of gross hacking keel condos habits the idea is the following in ghkk the mirror algebra is built from the scattering diagram and the scattering diagram is built by specifying the initial walls and using the condos habits soba man algorithm therefore for comparison with ghkk by the uniqueness property of the condos soba man algorithm it suffices to compare the set of incoming walls however we have defined walls simply as images of tweaks of tropical curves and we do not have a notion of incoming or outgoing walls in this generality therefore here in the cluster case we need to introduce a more restrictive notion of tweaks and walls which will allow us to distinguish incoming versus outgoing walls and to better control the monomials in the scattering functions we will call them c-tweaks and c-walls where c stands for cluster so a c-tweak is a tweak such that each infinite leg maps to a hyperplane e-prop with derivative sum multiple of e for sum e in this basis corresponding to unfrozen variables it's easy to see that for all stable map in the modular space m smooth that we are interested in the tweaks of the tropical curve associated to such stable map they are all c-tweaks in the cluster case and the next we define a c-wall to be a pair sigma n where n lies in p the sub monoid of m generated by s and the sigma in the hyperplane m-prop is a closed convex rational polyhedral cone a c-wall is called incoming if n lies in sigma and is called outgoing otherwise so c-wall has a little bit more refined information of this direction vector n and now we can construct a collection of c-walls by induction so we start with w0 the collection of c-walls of form e-prop n where e is any basis vector corresponding to unfrozen variable and n is any positive multiple of e we call these the initial c-walls and then by induction we define assume we already have w0 until wt and we define wt plus 1 as follows for all pairs for two c-walls all pairs of c-walls in wt such that either the pairing of the two direction vectors is nonzero or the direction vectors are parallel we define the sum of the c-walls of the two c-walls like this where the support of the wall is sigma 1 intersects sigma 2 minus all positive multiples of n1 plus n2 and the direction of the c-wall is just n1 plus m2 we check that it is a c-wall and we add all such sums to wt and finally we let w be the union of all wt so here it's important that we do not add walls when the two direction vectors as pairing zero but they are non-parallel because this corresponds to a non-transverse situation so two consequences of the construction are the following first the incoming c-walls of w are exactly the initial c-walls and the second for any generic c-tweak and any edge there is always some c-wall in our collection w such that the image of the edge lies in the support of the c-wall and the derivative is equal to the direction of the c-wall and we remark that for the scattering diagram associated to you the second consequence implies that the wall crossing function always has the form has this form where the exponent is just the multiples of this n whose prep contains x so a priori the exponents can be quite arbitrary but from the notions of c-tweaks and the c-walls we see that we have a strong restriction on the shape of the scattering functions now we are ready to deduce the comparison with ghkk so let the dghkk be the scattering diagram in ghkk it is produced by the condensate stubborn algorithm from the set of initial walls e-prop with scattering function one plus z to the e for all e for all basis vector e corresponding to the unfrozen variables therefore by the uniqueness property of the condensate stubborn algorithm and by the identification of incoming c-walls in our setting just above it remains to show that our c-walls have the same wall crossing functions and in other words it's enough to show the following claim that for each e in s and for any point x in e-prop generic the attached scattering function for getting curve classes is just one plus z to the e and that can be done by first figuring out what do the tweaks look like and then by an explicit computation so from this we deduce immediately the comparison theorem for the scattering diagrams between our scattering diagram and the ghkk scattering diagram and then we have the comparison theorem in the both a cluster case and also the x cluster case so let us recall that in the first lecture I mentioned the five consequences of the comparison theorem and roughly the comparison theorem gives geometric interpretations of and also more conceptual understandings of many constructions in ghkk and also proves some of their conjectures since I don't have much time for in the first lecture I mentioned another application of our theory to the study of for modular spaces of calabiol pairs so probably I can explain it in some other occasions thank you very much for your attention okay thank you thank you rush we're lectures and maybe some people want to ask some questions just please just unmute yourself and ask questions yeah yeah no for me the last part was very familiar so it's too easy yeah maybe the general questions in the situation we don't have nice assumptions about a fineness when we don't have a fineness with the mirror algebra is not an algebra yeah it's just formal but still yeah it's still for it's still algebra some form post-heurism yes yes so probably you want it to be some affinoid algebra yeah yeah yeah without a fineness but in any case we need some positivity assumption yeah because as we have seen in the proof of the deformation invariance we want to prevent bubbles moving from the interior to the boundary yeah because if so we want to count analytic disks in in the log calabiol variety u we want we don't want to count analytic disks that has something to do with the boundary yeah and if we don't have any a fineness or other sort of positivity assumption then we cannot count analytic disks in u because it can have bubbles or it can deform and touch the boundary so i guess in order to be able to count analytic disks we need at least that the log calabiol to be proper over some affine yeah you need some kind of semi-positivity of the boundary yeah it's yeah it's actually very proper over affine yeah proper affine it's sufficient here but maybe it's but the proper over affine is equivalent to the condition that we have some positive we have some positive combination of boundary components which is nev but they feel good conditions if they both both conditions they are equivalent and i think it's yeah also such kind of variety appears in jt theory usually jt you produce i think that's a satisfactory setting yeah to assume proper over affine it also projective projective or fine yeah and it also covers just the projective case over a point okay so there's no more questions then we can thank you for the nice series of lectures and thank you Maxine and thank you everyone for your attentions
 Thank you for coming to the last session of this mini course.  Let me first share with you the files in the chat and also...  Yes, so let's start.  Here is where we were on Monday.  The idea of the gluing formula is as follows.  Roughly the gluing formula states that we can glue two open curves along two opposite  boundary components and form a bigger concatenated curve like this.  And the count of the concatenated curve should be the product of the counts of the two initial  curves by this formula.  So this is more convincing evidence that our counts really reflect open curve counting.  It is also an essential ingredient in the proof of the associativity of the mirror algebra.  Now let's establish this gluing formula in three steps.  Step one, given two spans S1 and S2 in the essential skeleton of U, assume we have an  infinite one-valent vertex W1 and an internal infinite one-valent vertex, meaning that it's  mapped to the essential skeleton instead of to the boundary.  So assume we have such a vertex W1 in S1 and respectively we have an internal infinite  one-valent vertex W2 in S2 such that the two vertices map to the same point in the essential  skeleton SKU.  Next we consider a span delta with three infinite one-valent vertices W1, W2, W and mapping  constantly to this point where the previous W1 and W2 go.  So we consider this span which maps constantly to the same point and this span has just one  three-valent vertex and three infinite legs. So we can glue this constant span delta to  S1 and S2 along W1 and W2 and form a new span S like this in the essential skeleton of  U.  Here the infinite one-valent vertices W1 and W2 they become nodes in the new span after  we glue. So in the new span this is an infinite edge containing a node and this is another  infinite edge containing a node.  Now we have the following M. For any curve class gamma we have the following equality  that is the count associated to the span S and the curve class gamma by evaluating at  this internal marked point W is equal to the sum over all the compositions of gamma into  gamma 1 plus gamma 2 of the count associated to the span S1 curve class gamma 1 times the  count associated to the span S2 curve class gamma 2.  So for S1 we evaluate at the point W1 and for S2 we evaluate at the point W2.  So this is the formula we have for the count of the glued span since this part is constant  it doesn't contribute.  The proof is not difficult by passing to a big enough base field extension this follows  from a set theoretical decomposition of the set of skeletal curves associated to S of  the left hand side to products of sets of skeletal curves associated to S1 and S2 respectively.  So after big enough base field extension it reduces just to a set theoretical equality  one can just check by hand.  So this is the first step of gluing we glue two spines with this auxiliary spine at these  infinite vertices.  Now let's consider the second step of gluing.  So in the second step we are given two spines S1 and S2 in the essential skeleton of U  both transverse to walls and assume we have a point P1 in gamma 1 the domain of S1 and  P2 in gamma 2 the domain of S2 both in the interior of some edge and we assume that they  map to the same point in the essential skeleton and also they do not meet wall.  Since they map to the same point we can glue so we can glue S1 and S2 along the points  P1 and P2 and obtain a new transverse spine S in the essential skeleton of U like this  we just glue these two together at the point P1 and P2.  Then we have a similar lemma saying that for any curve class gamma we have the following  equality which says that the count associated to the glued spine S and any curve class gamma  is equal to the sum over all the compositions of gamma into gamma 1 plus gamma 2 of the  count associated to the first spine S1 and the curve class gamma 1 times the count associated  to the second spine S2 and the curve class gamma 2.  Here is the proof for this for this equality so for the proof we add an infinite leg W  to S at P because if you recall that in our definition of counts we always need some internal  mark point in order to evaluate and we proved using skeletal curves we proved a symmetry  theorem saying that the place where we evaluate eventually does not matter but for the just  the first to construct the count we always need some internal mark point so let's add  it let's add the infinite leg W to S at P at this the point we glue P and then we count  this spine by evaluating at this point W and after adding this infinite leg now we can  deform by stretching this point P and make appear two small edges E1 and E2 so this is  a small deformation of this by stretching the point P and that now W is attached the  leg W is attached to the middle of this purple of the two new purple edges and if we further  stretch if we further stretch the two vertical edges E1 and E2 to make them of infinite length  and contains contain a node so we make each edge infinite length to contain a node then  we arrive at the gluing situation of step one and note that in this stretching process  all spines are transverse so their counts do not change by deformation invariance and  now we just conclude by the lemma we proved in step one we show and we conclude our proof  for this equality so this is the second step to establish the gluing formula it's it uses  a deformation invariance for transverse spines as well as the previous lemma that we established  for the counts of such a glued span with this auxiliary with this auxiliary span delta in  the middle so finally in step three for the gluing formula we are given two spines S1  S2 in the essential skeleton of you both transverse to wall like this S1 S2 and assume we have  finite one valent vertices V1 in the domain gamma 1 of S1 and the V2 in the domain gamma  2 of S2 such that they map to the same point in the essential skeleton of you and we assume  that we have opposite derivatives meaning that the derivative at V1 of H1 plus the derivative  at V2 of H2 is zero since they have opposite derivatives and they map to the same point  in the essential skeleton of you we can glue S1 and S2 along V1 and V2 and obtain a new  transverse span S like this in the essential skeleton and here is the final theorem the  final gluing formula for any curve class gamma we have the following equality which says that  the count associated to this glued span S and any curve class gamma is equal to the sum of  all the over all the compositions of gamma into gamma 1 plus gamma 2 of the count associated to  the left part S1 with curve class gamma 1 times the count associated to S2 and the curve class  gamma 2 so this is the final gluing formula and it is a generalization of the two dimensional  case in my previous paper but here I'm presenting a more conceptual proof via deformation invariance  so the proof the idea of the proof is the following so we want to prove that when we  glue these two together we have this formula for the count and let's make a small extension of S1  this is our S1 we make a small extension of our S1 at V1 to S1 hat by linearity so we just extend  linearly at this vertex V1 extend a little bit this purple part is our extension and similarly we  make a small extension of S2 at V2 by linearity to S2 hat and by deformation invariance for this  transverse truncated spice the count remain the same the counts do not change when we make these  small extensions as long as we do not meet walls and now after making the two small extensions  observe that if we glue S1 hat and S2 hat together by identifying V1 and V2 so let's glue S1 hat  and S2 hat together at V1 and V2 we see that once we do that this gluing is actually just equal to  the glued span S to the gluing of S with some small straight span L so this gluing S1 hat S2  hat at V1 and V2 it's just it's equal to the gluing of S this red S together with this purple  purple edge L at the point V and both sides of this formula are gluing of two spans so we can  so both sides are gluing of other such gluing situation of step two so we can apply step two  to both sides of the above equality and we obtain this which says that the sum over all the  compositions of gamma into gamma 1 and gamma 2 of the count associated to S1 hat and the  gamma 1 times the count associated to this extended span this small extension S2 hat and  gamma 2 is equal to the sum over all the compositions of gamma into beta 1 plus beta 2 of the count  associated to S this red S and the curve class beta 1 times the count associated to the L this  purple small straight span L and curve class beta 2 so now we have to let's compute explicitly the  contribution of this part so we can explicitly compute that the count associated to L and the  beta 2 is equal to 1 if beta 2 is 0 and it's equal to 0 otherwise and if we substitute if we  substitute this explicit computation into the equality above we obtain the gluing formula  in R0 sorry do you understand you have this purple interval L maps to a point is essentially the  only contribution you have yeah L doesn't map to a point L map L maps to a small interval a small  interval I see yeah yeah so it's really yeah it's not clear what does correspond to in  simpleclic topology because you don't consider p1s but kind of like annular yeah it's yeah it's  a very small annular map into a place without walls I see any walls I see yeah so that place is  like just a torus so we have it's like we count annulus just in an algebraic torus yes so the  count is really the simplest yeah either one if a curve class is zero or zero if it has if we put  some non trivial curve class because it it's just a maps to a place without walls so there is nothing  interesting happening yeah so via these three steps we obtain this gluing formula zero and  let me remark that similar idea can be applied to show that our accounts are independent on the  choice of the torus when we impose the toric tail condition assume we have two torus embeddings  tm in you and the tm prime in you just two different embeddings leading to two different  toric tail conditions t and t prime so recall that in the definition of our naive accounts of  skeletal curves in order to obtain a finite dimension or modular space we need to impose  some extra condition and the some extra regularity condition on the boundary via analytic  continuation and that the condition called the toric tail condition was formulated according to  the choice of some torus torus embedding now we want to show that it's independent of the  torus embedding using the same idea in the proof of the gluing formula so assume we have two  torus embeddings tm in you and the tm prime in you leading to two different the toric tail conditions  quality and quality prime and now we consider a span s like this in the essential skeleton of you  with a finite one valent vertex v so here the span s is the whole graph including the purple part  this is our span s it has a finite one valent vertex v so we want to show that the count of  this span s does not depend on which tail condition we impose at this end v for this let's pick w  some point w very close to v and let l denote the restriction of the span s to this small interval  w v so this purple interval is just a restriction of s to a small neighborhood of v and then we pick  any point x in the middle of w and v and we consider the gluing of s with l so it's similar to the  gluing we considered a moment ago here when we consider gluing of s and l we get like this  part gets doubled it's like they're double now by step two above we obtain the following equality  so here you see that we are gluing we are gluing two spans at some interior point of edges which  is the situation of step two and so we can apply the formula in step two and now if we apply the  formula in step two we obtain the following so the left hand side yeah so when we apply we think  like this for the the left hand side is the sum over all the compositions of gamma into beta plus  delta of the count associated to s using the tail condition the first the tail condition t everywhere  times the count associated to l using the tail condition t prime at v and t at all at the other  end of l so this left hand side so we can think of l as for the left hand side we put the toric  tail condition t everywhere on s but for l we put the toric tail condition t at w but tail  condition t prime at v and then we apply the step two above to this gluing now for the right hand  side we just think that for the gluing we switch we switch this part of l with this part of s which  means that it's the sum over all the compositions of gamma into beta plus delta of the count  associated to our span s where we use the tail condition t prime at v and t in all other places  now at times the count associated to this small interval l where we apply tail condition t at  both ends so it's really so the left hand side as I said the difference between the left hand  side and the right hand side it's how we think of the gluing it's like we are switching that half  of l with that last piece of s so one piece has the toric tail condition t attached and the other  piece has toric tail condition t prime attached when we switch them in this gluing we obtain and  when we apply step two we obtain this we obtain this formula then this equality so now it remains to  compute explicitly the computations of the small this l piece so we can compute as before that the  count associated to l using the tail condition t and the curve class delta is equal to one if delta  is zero and if it's equal to zero for all other delta and the same holds when we count the small  interval l using toric using tail condition t prime at v and t at w the same equality the same count  works also for the other one and we now we substitute we substitute this explicit computation into the  equality above we obtain the following theorem for tail condition with varying torus so we have  proved by substituting this explicit computation into the equality we have proved that the count  associated to our spine s using tail condition t everywhere and and for some fixed class gamma  is equal to the count associated to the spine s equal to the count associated to the spine s  using tail condition t prime at v and tail condition t everywhere else these two counts are equal  so here we are just switching tail condition at one vertex then of course one can switch at all  other vertices and we can even apply different toric tail conditions at all finite vertices  so this in other words the theorem shows that the count of skeletal curves is independent  is independent is independent on the choice of the torus tm inside u  so just a small remark concerning the computation for the count associated to this small interval  where we put different toric tail conditions at both ends so the explicit computation for this  count using two different tail conditions is a bit more subtle than for the count where we put  the same tail conditions we need to use a result concerning the gluing of non-archimede  polyannual in my previous paper if we put two different tail conditions because if we put the  same tail condition then the count is easy it's like something yes sir you know maybe you mean  that on one part of interval every put condition t prime another t but your notation your notation  don't say this it's kind of like t prime everywhere yeah yeah but actually it says that it's one end  is t the other end is t prime i see so i hope that it's understandable okay otherwise the notation  is a bit complicated yeah so i'm saying that if at both ends we put the same tail condition  then the count is easy to do because yeah yeah as maxi remarked we just map to a torus so then  it's just counting something in the toric variety but if we put different tails at the two ends  then the counting is a bit more complicated because it maps to some gluing of gluing of two  different torus but one can show that actually one can decompose the the automorphism of the some  annulus into two parts where one part can be extend can be extended to a automorphism of  some disk times poly annulus and another part can be extended to automorphism of another  opposite disk times the annulus so after applying this automorphism we see that we are we we are back  to the previous situation just counting something in p1 times some poly annulus the computation of  automorphisms of this automorphisms of the some affinoid algebra  yes yeah so this is the idea for the proof of the gluing formula and also for the proof of  changing tail condition and in fact this proof of changing tail condition  also works if we eventually use more general tail conditions without using any embedded torus  so  it's easier to carry out this proof than one can imagine using like deformation variance  for it feels a bit more complicated so now let's turn to the next section  structure constants and associativity of mirror algebra  we want to apply our gluing formula and all the other techniques we have developed so far  to study structure constants and the associativity of the mirror algebra  so let us recall the setting from the first lecture where we have log kala b y  of variety u containing some torus tm and we have some s and c compactification of u in y  we have monoid ring r over this effective curve classes assembling all possible curve classes  together and the mirror algebra a it is it has basis the integer points in the essential  skeleton as an arm module so this was our setup in the first lecture and again recall that given  some integer points in the skeleton we write the product in the mirror algebra a as  the this product of the theta functions theta p1 to theta pn in the mirror algebra a as this sum  so first we sum over all integer points in the essential skeleton of the basis vector theta q  and then we sum over all curve classes over all effective curve classes of the basis element z  to the gamma it's just a notation for the basis elements and we denote the coefficient by  chi p1 to pn q gamma and this chi is called the structure constants of our mirror algebra a  and let's also recall how it was defined in in the first lecture  the structure constant chi was defined as follows we had so first we figure out what was the class  delta of the editory tail and we denote the total class beta to be gamma plus delta beta is supposed  to be the class of the the closed p1 after the total tail extension we also had the z  equal to the opposite of q in the essential skeleton and we had topo pz  a putting p p1 to pn and z together and we use this topo pz to specify intersection numbers  with the boundary  then we considered the modular space this hpz beta with marked points labeled as  p1 to pn zs so this is the modular space of maps of maps from p1 with marked points p1 to pn zs  and we say that we specify the intersection of the p1  so it's the modular space of maps of p1 into y and we specify the intersection numbers  at these marked points with the boundary d using the topo pz  and we had a natural map phi from this modular space to here taking domain  taking domain and evaluation at the last marked point s and we also had a special point qtodah  in the identification of the target which was given as a pair of mu and q  mu we give mu we just specify what was mu in the first factor  corresponds to some divisory evaluation and q is just the given q  so the given q is the integer point in the essential skeleton in particular it's a point  in the identification of u so we had this special point qtodah and then  finally we had a subspace f in the fiber of the map phi over qtodah which is a finite analytic space  and if we take its length the length was by definition the structure constant kind  of a  in the first lecture first I give a heuristic picture of what we count for the structure constant  about counting disks with some conditions on the derivative of the disk at the boundary and  and after that I give a precise construction of the structure constant in using algebraic using  noic median geometry so this was just a recall of what we did and the definition is quite straightforward  but let's remark the following  due to the choice of this specific point qtodah inside the target so we used this qtodah  to take fiber and then take subspace due to the choice of this specific point the curves in f  are responsible for structure constants although highly generic in the algebraic sense are in fact  very special in other words non-transverse from the tropical viewpoint because this special point  it's a generic point in the algebraic world but it's a very non-generic point in the tropical world  so this results  very generic curves in the algebraic sense but very special curves from the tropical in the  tropical picture and this was convenient for giving a quick definition of structure constants  but it's impractical for proving any properties about them for example associativity  finiteness all these properties they are out of reach from this quick definition  we must deform the curves in f into more transverse positions by perturbing this special point qtodah  because when we want to prove properties for example if we want to apply the gluing formula  or deformation invariance we usually need the assumption that the spines are transverse  but if we do the count using this specific point qtodah we will not get transverse spines  so we must perturb these curves by varying the point qtodah do we do it like this  for a position we label the marked points of metric trees in this modular space  so this is a modular space of tropical curves rational tropical curves with n plus one n plus two legs  here abstract tropical curves are the same as metric trees and we label the n plus two points  marked points as p1 to pn zs and let vm in the modular space be the subset consisting of metric  trees whose z leg and s leg are incident to a single three valent vertex here is the picture  we have such a metric tree it has a lot of legs by leg we mean infinite one valent vertex or yeah  so or more precisely we mean these edges containing infinite one valent vertices so here we consider  the subset where the z leg and the s leg they are incident to a single three valent vertex  so the z leg and the s leg they meet first and before the tree branches to other legs  yeah and we observe that this vm is is a neighborhood of this special choice of the  modulus mu and next next we want to figure out a neighborhood of this special point q  so let's consider a polyhedral subdivision sigma of the essential skeleton given by the set of  walls in the essential skeleton here we can assume the set of walls to be finite polyhedral by bounding  the degree of twigs by the fixed curve class beta so in general uh wall the set of walls they are  infinitely many walls and it can be dense in the essential skeleton but if we bound some degree  we get a finite set of walls so we consider the polyhedral subdivision induced by the set of walls  and we let vq be the open star of the point q in sigma in other words the union of open cells  in sigma whose closure contains q so we have q and we have a polyhedral decomposition we just  take a cone around around the point q q might be q might lie inside a wall but it doesn't matter  we take the star of q in sigma then q becomes an interior point in this open star  and we said that by construction q tilde which was given as a pair mu q mu is the interior  vm is a neighborhood of mu vq is enabled of q so q tilde so vm times vq is a neighborhood of q  tilde inside the product of the tropical modular space and the essential skeleton  and we already remarked when we talk about essential skeletons that using Tamkin's  matrization theory one can show that product of skeleton is homomorphic to skeleton of product  and this lies in the identification of the product  yeah so using this construction we figure we figure out two natural neighborhood of sorry  we figure out a natural neighborhood of the first factor of q tilde and a natural neighborhood of  the second factor of q tilde so we figure out a neighborhood of q tilde i recall that our goal  is to perturb the point q tilde so we will be perturbing q tilde inside this neighborhood vm  times vq and now let this be the pre-image of vm times vq by the map phi analytic from this modular  space by taking domain and evaluation at the last mark point s so before we took pre-image of a  single point q tilde now we take pre-image of this neighborhood of q tilde and now we take  curly f to be the subset in the pre-image satisfying the toric tail condition then the  preposition says that phi analytic is finite at all on the neighborhood of this subset  and whose degree gives the structure constant so this is how we perturb the special point  q tilde into general position by allowing q tilde to vary inside this neighborhood vm times vq and  we prove that if we allow q tilde to vary and when we impose toric tail condition then this map phi  is still is good it's finite at all on the neighborhood of this f so the degree is well  defined and it gives the structure constant since it's finite at all so we get a well defined degree  and the the degree at the fiber over the point q tilde it was our quick definition of the structure  constant so here the structure constant is reinterpreted as some degree of finite at our map  and for its proof we use the toric tail preposition for almost the transverse  spines because here it feels like deformation invariance that we are moving we are moving  the spines for the structure constants inside this conical neighborhood but  they do not state it's not always a transverse sometimes it it becomes non-transverse especially  at the point of q tilde for example but we had developed this reposition in the last lecture  not just for transverse spines but is that it is specially adapted to the situation here where we  can go across the walls so after this perturbation into generic position now we can prove the  following theorem the multiplication rule given by the structure constants chi here is commutative  and associative here is the the sketch of proof commutativity is obvious because the definition  of the structure constant chi is symmetric with respect to the p i's and associativity means  that the product theta p1 theta p2 to theta pn does not change if we add arbitrary parentheses  so let us now sketch the proof of the following equality where the left hand side means we first  take product of theta p1 and the theta p2 and then we take product with theta p3 while the right  hand side means we take the product theta p1 theta p2 theta p3 together using the multiplication rule  so we want to prove this equality and we just rewrite the products using the multiplication rule  and we substitute the multiplication rule into the equality and we see that the equality becomes  equivalent to the following equality for every integer point q in the essential skeleton and  every curve class gamma now observe that the right hand side of the equality star is given by  accounts of skeletal curves associated to the spines of this shape where we have three infinite  lags with derivatives p1 p2 p3 respectively and also we have one finite lag with the inward  derivative q so three infinite lags with outward derivative p1 p2 p3 and one finite lag with  inward derivative q and by the above proposition about perturbing q tilde we can deform the modulus  of the domain here by stretching this point so we deform this point into a small path l and then  we further stretch this path l very very long like this so if we stretch the path l very long  we see that the point u near the top of the path l will map sufficiently close to the ray  zero r inside the essential skeleton of u and we need it to be sufficiently close to the ray  because we want it to lie inside the cone vr in the essential skeleton of u in as in the above  proposition for the structure constant chi p1 p2 r eta recall that in our quick definition of  structure constants we say that the marked point just go directly to the point q tilde  and that was two non-transverse two rigid now we allow q tilde to move a little bit around but we  still we always need it to be sufficiently close to the ray oq so it it should not move outside  across some walls around this ray otherwise it doesn't give the correct structure constant  so here we stretch this path l very long so that finally some point u near the top of the path  will be sufficiently close to the ray or then then that will be good enough for defining  the structure constant chi p1 p2 r eta so we can cut at the cross u apply the gluing formula  and obtain the left hand side of the equality star so similarly given two spines responsible  for the product in the left hand of the equality star we can glue them to form a highly stretched  spine like this that is responsible for the right hand side and this completes the proof of associativity  so before the break let me quickly sketch another important property of our structure constants  which is the convexity property so here is the theorem of the convexity property  let f be a Cartier divisor on y and consider consider an analytic disc  in general position responsible for the structure constant for the structure constant chi p1 to pn  q gamma then the following hold first since f is a Cartier divisor we can take its tropicalization  and obtain a real valued function f chop on the essential skeleton  then first we have that the sum of f chop at overall pi minus f chop at q is equal to the  intersection number between f and gamma minus the degree of f and liitified restricted to the  punctured disc meaning the disc minus all the marked points and the second if f is an f and  the minus f restricted to you is effective then we have f chop of q is less than or equal to the  sum of f chop at every pi and furthermore assume f is ample and the minus f restricted to you is  effective then the above equality inequality is an equality if and only if f maps the punctured disc  into the torus and the proof uses some detailed computation using semi-stable models of curves  the convexity theorem implies the following finiteness result finiteness result one given  p1 to pn in the essential skeleton of you some integer points in the essential skeleton of you  then there are at most finitely many pairs q gamma where q is an integer point in the essential  skeleton of you and the gamma is a curve class such that the structure constant is nonzero  let me give a quick proof how the finiteness follows from the convexity  so since u is a fine we can find the regular functions x1 to xl on you such that the set  of points in the essential skeleton where the norm of xi at b is bounded by some real number c  for all i this set is a is bounded for any real number c now if the structure constant is nonzero  so by the convexity statement two we apply the convexity statement to the car t a divisor given  by these regular functions we obtain this equality the norm of xi of the function xi at q is less  than or equal to a sum of is less than or equal to sum over all j of x the norm of xi at pj  and this shows that given p1 to pn there are at most finitely many q such that the structure  constant is nonzero for some gamma so this bounds q and the next let's bound the gamma  we want to bound both q and the gamma and the assumption that u is a fine implies that there  is an ample divisor f on y such that minus f restricted to u is effective  and now we apply the convexity theorem one the statement one this statement and we obtain  this the following equal the following equality and inequality  which says that the intersection number between f and the gamma is equal to the sum of f drop  at every pi minus f drop at q plus the degree of the analytic f restricted to the punctured disk  and since minus f restricted to u is effective and this degree is non-positive so we see that  this is less than equal to this and this is fixed the right hand side this bounds gamma by the  amponeness of f so this is how we deduce the finiteness result from the convexity property  and the finiteness result is important because it implies that the two sums in the multiplication  rule here they are finite sums so the multiplication rule gives an algebra structure on the free r  module a instead of just some formal algebra structure and in fact  we have the following stronger finiteness result too which says that the mirror algebra  is a finitely generated r algebra and for its proof we need to result to the  equivalent boundary torus action on the mirror algebra so here due to time constraints I will  omit this boundary torus action and finite generation result in this lecture and after the  break I will explain the application towards cluster algebra and also I will explain the  wall crossing how to get scattering diagrams using these counts of analytic curves so let's make  five minutes break okay thank you so here is the plan for the last part of this lecture  first I'll explain how to construct a scattering diagram via infinitesimal analytic cylinders  and the second I'll prove the property of theta function consistency for the scattering diagram  and third we will set all curve classes to zero so that we no longer care about the  compactification and the fourth I'll explain the class we will apply the above to the case of  cluster algebra where I need to introduce two new notions C twigs and the C walls especially for the  cluster case and finally I'll explain the comparison with the work of gross hacking  kio condo savage on the for class with the work of gross hacking kio condo savage for cluster  algebras so let's start with first scattering diagram via infinitesimal analytic cylinders  both in the yes  both in the original suggestions by condos savage soba man and in the gross seabird program  the construction of a mirror variety relies on the combinatorial algorithmic construction  of scattering diagram also known as wall crossing structure our construction our construction of  the mirror algebra by counting non-archimedean analytic disks as in the previous lectures  completely bypasses any use of scattering diagram nevertheless our geometric approach also allows  us to give a direct construction of the scattering diagram by counting infinitesimal analytic  cylinders without the step-by-step condos savage soba man algorithm and this has three implications  first it gives a geometric interpretation of the combinatorial scattering diagram  second conversely we obtain a combinatorial way for computing the non-archimedean curve counts  and third it paves the way for the comparison with the work of gross hacking kio condo savage on  cluster algebras let me also remark that there are recent works of argus gross gross seabird  gross seabird give another geometric interpretation of scattering diagram based on the theory of  punctured log curves developed developed by abramovich chain gross and seabird  now let us sketch our construction of the scattering diagram via infinitesimal analytic cylinders  recall we have our log halabi au containing some torus tm and is contained in some s and c  compactification y and we denote by n the dual of m  so definition given a hyperplane n-purple in mr m tensor with r and any generic point x in the  hyperplane m-purple generic means that it's not contained in any other n-purple it's only contained  in this one hyperplane and say we are given two vectors v and w in m minus n-purp and the curve  class alpha let v x v w be the infinitesimal spine here bending at x with incoming direction w  and outgoing direction v and this gives rise to the associated count of analytic curves  n v alpha so we consider this infinitesimal spine bending once at a generic point x in the  hyperplane m-purple with specified the incoming and outgoing direction and we consider the associated  count of analytical curves then using all these counts for any x in some hyper for any generic  point x in some hyperplane and perp we define the following wall crossing transformation  and psi x n acts on the basis vector z to the v as follows if we if so for all v in m  which pairs with n positively we define the value of psi x n at z to the v to be  the sum over all possible vector of all vectors of every vector w in m which pairs with n positively  and over all curve class of the basis vector z alpha z w with coefficient the count we just defined  so in other words we just sum over all possible incoming direction w and all possible curve class  and if for all v in m which pairs with n trivially which lies in the  hyperplane we just defined it to be z to the v it doesn't the wall crossing transformation  does nothing for z v and unlike in the multiplication rule this sum does not is not a finite sum  but this converges in the natural eddyc topology which we describe here  since this cone of curves may not be polyhedral we fix a strictly convex toric monoid q containing  this effective curve classes and let our hat be the completion of the monoid ring  over q direct sum m with respect to the maximum monomial ideal i in other words the ideal generated  by monomials z qz m with q nonzero and arbitrary m so we use this completion to express the convergence  lemma convergence lemma the formal sum in the wall crossing transformation lies in our hat  so the reason is that when we give bound on curve classes it implies bound on the combinatorial  types of the two-weeks of all analytical curves contributing to these counts and this in turn  gives a bound on the incoming directions w so now by linearity over this monoid ring we can extend  the wall crossing transformation transformation to a map from the monoid ring over all curve classes  plus direct sum with the subset of the lattice m which pairs with m nonactively and the map  goes to our hat we just here we defined what it does on the basis vectors and we extended  by linearity over the monoid ring theorem wall crossing homomorphism theorem the map psi xn is a ring  homomorphism so it means that if we pick arbitrary m1 m2 in m that pairs with n nonactively we want  to show that we have a psi applied to z to the power n1 plus m2 is equal to psi of z to the m1  times psi to the z of m2 it's what means for the map to be a ring homomorphism  so let's try to prove this we let's fix any vector e in m and any curve class alpha and  let's prove the equality of the coefficients of z alpha ze inside the formula above  note that it's obvious the equality is obvious when both m1 and m2 lies in n-perf  because if they lie in they lie in n-perf then the wall crossing transformation does nothing on them  so we may assume that one of them pairs with n positively so assume that the pairing between n  and m1 is positive and the pairing between n and m2 is non-strictly positive and we consider the  count of analytic pair of pants associated to this spines with three just various infinitesimal  spine with three legs with three ends near x with directions m1 m2 minus e at the three ends so claim  suppose x is contained in the wall sigma  then the count of analytic pair of pants is independent of which side of sigma the three  valent vertex of this our spine maps to here the left picture shows  this spine mapping to the essential skeleton and this blue blue line is a wall and the left picture  shows the situation where the three valent vertex maps to the left the three valent vertex maps to  the left and the right picture shows the situation where the three valent vertex maps to the right  so the claim says that the count is independent of which side the three valent vertex go  let's further observe that in the left picture when we specify the directions m1 m2 and e  this the shape of the spine is unique which we denote by sl because there is only one possible  band and that band is determined by the three directions at the ends however in the right  picture the shape of the spine is not unique because we have two bands and  this band when we deform from left to right this band is decomposed into two bands and all  possible ways of decomposition are allowed so for the right hand picture we are actually  in summing over many we are summing over many different shapes of spines which we denote by sri  sorry Tony yeah the source stability is even m2 is parallel to the wall which is yes yes this is  the trickiest this is the trickiest possibility but i allow m2 to be parallel to the wall here  what is your doubt where is your doubt no doubt no no it's allowed yeah yeah and it's exactly  the possibility where m2 is parallel to the wall that possibility implies that like the  that the preservation of volume elements we see so yeah so we keep in mind that we also have  the possibility m2 parallel to the wall yeah so as Maxim remarked that here in the proof we use  the proposition for toric for tail conditions in families in the last lecture and the trickiest  case is when one of m1 or m2 lies in m-purp one of them is parallel to the wall where we need to use  the deformation invariance for almost the transverse suspense  so now let's go back to the proof of the theorem this is proof for the claim about  deformation invariance when we move from left to right and now let's back to the proof of the  wall crossing homomorphism theorem note that for any span l disjoint from walls the count  nl gamma is 1 if gamma is 0 and is 0 otherwise because if it's disjoint from walls we are  essentially in the toric situation and everything can be computed explicitly  keeping this in mind now we consider first the left picture if we cut the spine sl at the cross  by the grouin formula then by the grouin formula the count  associated to s alpha sl this red span sl and any curve class alpha gives the coefficient of  z alpha ze in this wall crossing transformation because when we cut at this cross  the left part is disjoint from wall so it doesn't contribute while the right part has a band and  the right part is exactly the infinitesimal cylinders that we use in the definition of the  wall crossing formula so therefore when we count analytic curves associated to sl we get the coefficient  in this wall crossing transformation and the next if we cut next let's consider the right picture  where the three valent vertex maps to the right if we cut the spine sri at the two crosses but  then by the grouin formula so we cut at these two places now the spine is broken into three pieces  note that the right piece is disjoint from any walls so the right piece doesn't contribute  but the two left pieces they both contribute and they both have a band and we just add  their contributions together so by the grouin formula we see that the count associated to  this the count associated to this spine sri is equal to the sum over all the compositions of  curve class alpha into alpha one plus alpha two of the count associated to this upper left  spine times the count associated to the lower left part and we remarked that when the three valent  vertex maps to the right of the wall we have different shapes parametrized by i and next  we sum over all possible shapes sri and we see that the sum of counts associated to sri gives  exactly the coefficient of z alpha ze inside the product of the two wall crossing transformations  so we concluded the proof of wall crossing homomorphism by the claim yeah so this shows  that wall crossing transformation is a ring homomorphism and we remark that the immediate  consequence of the ring homomorphism is the following given any x generic any generic x in some  hyperplane n perp and any vector v in m which who's pairing with n equals one so in particular  this means that n is necessarily primitive vector we just rewrite psi xn at zv to be zv times some  function this is just a rewriting with some function in our head then the fact that wall crossing  is a ring homomorphism implies that the function f x and v does not depend on v  geometrically this means that the count of infinitesimal analytic cylinders depends only on  the amount of band independent of incoming or outgoing direction so for the count of this  we can change incoming or outgoing direction as long as we have the same amount of band  we always have the same count in particular this implies that the wall crossing transformation  preserves the standard volume form on tori and moreover we have  we have we have the equality between the function fx and v and fx minus and v  in other words it's independent of the orientation of wall the function is independent of orientation  of wall so we can denote fx so we can denote fx and v just by fx since it's independent of n  and also of v for any choice of n primitive whose prep contains containing x and any choice of v  with the and the pairing between nv equals one and we call fx the wall crossing function attached to x  so here is the conclusion now we can write the wall crossing transformation applied to  zv simply as zv times fx to the power the pairing between n and v for any  for any vector v in m that pairs with n non-actively and this is  the it resembles like to the more classical wall crossing  formula and using this formula we see that the wall crossing transformation  psi xn extends to an automorphism of the fraction field of our hat  let me remark that this may not give an automorphism of our hat since the wall crossing function  need not to be invertible in our hat because of curve classes  it might not start with one it might just start with a curve class and that curve class doesn't  is not invertible it doesn't matter first this is just how it is if we take into account curve  classes and it will become invertible when we set all curve classes to zero  now definition now we have all the wall crossing transformations  wall crossing functions we can define what is our scattering diagram so let d be the set of pairs  xfx where x is any generic point in m-perp in the hyperplane n-perp for any non-zero n  and fx is the associated wall crossing function we call this set of pairs xfx we call it the  the the scattering diagram associated to u with respect to the compactification y and also with  respect to the torus tm so scattering diagram depends on the torus and we remarked that by  the eddy convergence of fx of the wall crossing function if we mod ik where i was if we mod out  some power of i where i was the maximum monomial ideal then we obtain a finite scattering diagram  dk in other words finite means that we will only have finitely many polyhedral walls in dk here  in d we have infinitely many walls so it no longer makes sense to say the shape of each wall  they are so small it it's better just to give the wall crossing functions attached to each  generic point but once we mod out some power of the maximum monomial ideal we get finite  scattering diagrams dk with finitely many polyhedral walls and we call dk the case or the approximation  of d and one important property for scattering diagram is the consistency consistency property  so let's try to establish consistency  first let us establish a variant of consistency property called which we call theta function  consistency we introduce a new definition choose any generic point x in mr and the two  vectors me in m and let sp x me be the set of spines in mr with domain minus infinite infinity to zero  like this such that minus infinity maps to the boundary with derivative minus m  and zero maps to x with derivative minus e so we consider the set of all such red  spines starting from infinity ending at x with with directions m and e at the two ends and  this is related to the notion of broken line in ghkk we define the local theta function theta xm to  be the formal sum over all vector e and all such red spines s and all curve class of the vector z  alpha the basis vector z alpha ze and with the coefficient the count associated to the the red  span s and alpha and again we have addic convergence this lives in our head and we have the following  theta function consistency theorem the scattering diagram d is theta function consistent in the  following sense given any k we consider the case order approximation given any k we consider the  case order approximation of our scattering diagram dk and we take a polyhedral wall sigma with attached  wall crossing function f sigma inside dk and we choose a vector n such that the wall is contained  in the hyperplane and perp and we consider two points a and b two general points near  another general point x in the wall on two sides of the wall so we pick a general point x in the  wall and the two general points on two sides of the wall a and b near x such that n a pairs with  n positively and the b pairs with n negatively then we have the following if we apply the wall  crossing transformation per se to the local theta function at a with infinite direction m we obtain  the local theta function at b with the same infinite direction and conversely if we apply the wall  crossing transformation per se with respect to the opposite orientation of the wall to the local  theta function theta bm then we get back the local theta function theta a m this is what we call the  theta function consistency property and we use the gluing formula and the deformation in variance  for the proof so our next step is to forget all curve classes we want to get closer to the  classical wall crossing structure or classical scattering diagram without always thinking about  curve classes in our compactification so let's set all curve classes to zero  as I said our wall crossing transformation and the scattering diagram depends on the  compactification u inside y via the usage of curve classes in y so this is a more refined  information but we can remove this dependence by setting all curve classes to zero in order to have  attic convergence if we set all curve classes to zero we need to impose a condition on the  band of infinitesimal analytic cylinders without extra condition we will lose attic convergence  if we forget the curve classes the assumption is the following for any  for any non-zero count associated to some infinitesimal spine at x with the incoming direction  outgoing directions v and w if the count is non-zero then the band w minus v lies in the  strictly convex monoid p inside m we need this assumption in order to have attic convergence  when when we forget curve classes so and here is the new attic convergence we consider  thanks to the monoid m we consider j the maximum monoid ideal in the monoid ring over p  the monoid p and let l not hat be the genetic completion of zp and finally let l hat be l  not hat tensor with zm over zp in other words we are taking completion of zm in the direction of  the strictly convex monoid p we allow infinite sums in the p direction and only finite sums  in the other direction in all other directions so for now we will be forgetting curve classes  for a position under the quotient from the monoid ring associated to q direct sum p to the  monoid ring associated to p when we ignore all curve classes the wall crossing functions fx in r  they map to the quotient wall crossing functions fx bar in l hat meaning that when we ignore curve  classes they have attic convergence in l hat and the the wall crossing transformations  psi xn which used to be an automorphism of the fraction field of r hat they become  automorphisms of l hat so we no longer need to take a fraction field of l hat because when we  forget the curve classes the wall crossing transformations will automatically become invertible  and furthermore the local theta functions in r hat they map to the quotient theta functions  in l hat so we have this j attic convergence for all these things before and moreover we get  the invertibility of the wall crossing transformation  so now we denote du to be the set of pairs x fx bar where x is any generic point in any  hyperplane and probe and the fx bar is the associated wall crossing function ignoring all  curve classes and we call this the scattering diagram associated to u with respect to the torres tm  now we have the following consistency result  which says that the scattering diagram du is consistent in the sense of contrived  sobo man in other words for any general loop inside the mr the composition of wall crossing  automorphisms after ignoring curve classes along the loop is just identity  so for the proof we observe that the subring of l hat generated by all local theta functions  is genetically dense and then the theorem follows from the theta function consistency  so that's the general constructions of the scattering diagram using counts of infinitesimal  analytic cylinders and now let us apply our constructions above to the case of cluster  algebra now here is the cluster data we have lattice m with an integer valued skew symmetric form  and we have s prime a basis of m and s inside s prime a subset of s prime  this gives a seed for a skew symmetric cluster algebra of geometric type  where s corresponds to unfrozen variables and the s prime minus s corresponds to frozen variables  and the seed gives rise to curly a a fork contour of a type cluster variety  which is the gluing of torri via cluster mutations and we let a up b the algebra of  global functions on a called the upper cluster algebra  let us assume that the skew symmetric form is unimodular  we assume this for mainly for con partially for convenience  this holds in the principal coefficient case because this holds in the principal coefficient  case and then we will deduce more general cases from the principal coefficient case  and we also assume that the spec of the upper cluster algebra which we denote by u is a smooth  case so for example double brouha cells in semi-simple complex complex three groups  satisfy this smoothness assumption I think it's possible to extend our work also to cover the  non-smooth case without too much extra effort but I haven't checked all the details  then u is a log halabi all containing a torus dm so we can apply our theory to u and obtain  a mirror algebra as well as a canonical scattering diagram by counting non-archimede  and elliptic curves and here is the question how shall we compare with the constructions  in the paper of gross hacking keel condos habits  the idea is the following in ghkk the mirror algebra is built from the scattering diagram  and the scattering diagram is built by specifying the initial walls and using the condos habits  soba man algorithm therefore for comparison with ghkk by the uniqueness property of the condos  soba man algorithm it suffices to compare the set of incoming walls however we have defined  walls simply as images of tweaks of tropical curves and we do not have a notion of incoming  or outgoing walls in this generality therefore here in the cluster case we need to introduce a  more restrictive notion of tweaks and walls which will allow us to distinguish incoming  versus outgoing walls and to better control the monomials in the scattering functions  we will call them c-tweaks and c-walls where c stands for cluster  so a c-tweak is a tweak such that each infinite leg maps to a hyperplane e-prop  with derivative sum multiple of e for sum e in this basis corresponding to unfrozen variables  it's easy to see that for all stable map in the modular space m smooth that we are interested in  the tweaks of the tropical curve associated to such stable map they are all c-tweaks in the cluster  case and the next we define a c-wall to be a pair sigma n where n lies in p the sub monoid of m  generated by s and the sigma in the hyperplane m-prop is a closed convex rational polyhedral cone  a c-wall is called incoming if n lies in sigma and is called outgoing  otherwise so c-wall has a little bit more refined information of this direction vector n  and now we can construct a collection of c-walls by induction  so we start with w0 the collection of c-walls of form e-prop  n where e is any basis vector corresponding to unfrozen variable and n is any positive multiple  of e we call these the initial c-walls and then by induction we define assume we already have  w0 until wt and we define wt plus 1 as follows  for all pairs for two c-walls all pairs of c-walls in wt  such that either the pairing of the two direction vectors is nonzero or the direction vectors are  parallel we define the sum of the c-walls of the two c-walls like this where the support of the wall  is sigma 1 intersects sigma 2 minus all positive multiples of n1 plus n2 and the direction of  the c-wall is just n1 plus m2 we check that it is a c-wall and we add all such sums to wt  and finally we let w be the union of all wt so here it's important that we do not add walls  when the two direction vectors as pairing zero but they are non-parallel because this corresponds  to a non-transverse situation so two consequences of the construction are the following  first the incoming c-walls of w are exactly the initial c-walls and the second for any generic  c-tweak and any edge there is always some c-wall in our collection w such that the image of the edge  lies in the support of the c-wall and the derivative is equal to the direction of the c-wall  and we remark that for the scattering diagram associated to you the second consequence implies  that the wall crossing function always has the form has this form where the exponent is just  the multiples of this n whose prep contains x so a priori the exponents can be quite arbitrary but  from the notions of c-tweaks and the c-walls we see that we have a strong restriction on the  shape of the scattering functions now we are ready to deduce the comparison with ghkk  so let the dghkk be the scattering diagram in ghkk it is produced by the condensate  stubborn algorithm from the set of initial walls e-prop with scattering function one plus  z to the e for all e for all basis vector e corresponding to the unfrozen variables  therefore by the uniqueness property of the condensate stubborn algorithm and by the  identification of incoming c-walls in our setting just above it remains to show that our c-walls  have the same wall crossing functions and in other words it's enough to show the following claim  that for each e in s and for any point x in e-prop generic the attached scattering function  for getting curve classes is just one plus z to the e and that can be done by  first figuring out what do the tweaks look like and then by an explicit computation  so from this we deduce immediately the comparison theorem for the scattering diagrams  between our scattering diagram and the ghkk scattering diagram and then we have  the comparison theorem in the both a cluster case and also the x cluster case  so let us recall that in the first lecture I mentioned the five consequences of the comparison  theorem and roughly the comparison theorem gives geometric interpretations of and also  more conceptual understandings of many constructions in ghkk and also proves some of their conjectures  since I don't have much time for in the first lecture I mentioned another application of our  theory to the study of for modular spaces of calabiol pairs so probably I can explain it in  some other occasions thank you very much for your attention okay thank you thank you rush  we're lectures and maybe some people want to ask some questions just please just unmute yourself  and ask questions  yeah  yeah no for me the last part was very familiar so it's too easy yeah  maybe the general questions in the situation we don't have nice assumptions about a fineness  when we don't have a fineness with the mirror algebra is not an algebra yeah it's just  formal but still yeah it's still for it's still algebra some form post-heurism yes  yes so probably you want it to be some affinoid algebra yeah  yeah yeah without a fineness but in any case we need some positivity assumption  yeah because as we have seen in the proof of the deformation invariance we want to prevent  bubbles moving from the interior to the boundary  yeah  because if so we want to count analytic disks in in the log calabiol variety u  we want we don't want to count analytic disks that has something to do with the boundary  yeah and if we don't have any  a fineness or other sort of positivity assumption then we cannot count analytic disks in u  because it can have bubbles or it can deform and touch the boundary  so i guess in order to be able to count analytic disks we need at least that the log calabiol  to be proper over some affine yeah you need some kind of semi-positivity of the boundary yeah it's  yeah it's actually very proper over affine yeah proper affine it's sufficient here but maybe it's  but the proper over affine is equivalent to the condition that we have some  positive we have some positive combination of boundary components which is nev  but they feel good conditions if they both both conditions they are equivalent and  i think it's yeah also such kind of variety appears in jt theory usually jt you produce  i think that's a satisfactory setting yeah to assume proper over affine  it also projective projective or fine yeah and it also covers just the projective case over a point  okay so there's no more questions then we can  thank you for the nice series of lectures and thank you Maxine and thank you everyone for your  attentions
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