 pairs I'm going to call them smooth and F pairs These are pairs X and D and I will be writing S and P in short throughout my talk where X is a complex smooth projective variety of dimension M and D at least for my talk will be an anti-canonical divisor Which is effective reduced and normal crossings and I'm going to be writing all the way up to DR For some positive integer DR for it's a reducible components And I want to insist that each irreducible component DI is Moved so it has no negative intersection with any curve in in X examples So in dimension one I can take X to be the projective line and D to be the union of zero and infinity Just north and south the real sphere in this case M is equal to one and R is equal to two Or for one example two for Some more exciting action. I can take X to be the projective plane and D to be I Can choose if D is irreducible since I'm insisting that each irreducible component is smooth and D is anti-canonical Then D has to be a smooth cubic Or I can consider a two component case I can take D to be the union of a conic and The line that's not tangent to it or I could take D to be the union of the toric prime divisors of of the plane 1 2 3 of the coordinate axis Some piece of jargon that's not at all standard, but it's just something that I will employ For simplicity during my talk is that I'm gonna call a smooth and F pair X D Toric if X is and D is a Toric boundary T is the big Torus orbit in X and I'm gonna call it lohenga if X is a surface and D is singular so in particular since the irreducible case is always smooth This will lead to R being greater than one so example one is Toric The last example is Toric and lohenga The middle one is lohenga, but not Toric at a top one is neither Toric nor lohenga Have I butchered the Dutch pronunciation of the name enough? I'm asking for to the Dutchman in the audience Now the title that I've just raised is So basically the mantra of this talk is that I will want to relate a pair of enumerative theories of these pairs X and D so X a variety and D a divisor with special properties inside X and Well before I do it, let me forget for a second about the divisor all together and remind you about Gramm-Witton theory It's just a bit of a primer about rational from a Witton theory X So informally what I would like to do is if I have a collection of close-up varieties One to some number n inside X. I would like to attach to it some Count is informally the count of Degree D rational curves Passing through my given collection of cycles B Well the Gramm-Witton way of doing it is to consider parametrized curves So to X we attach the modular space M0 and bar Xd Which is the modular space of stable maps to X So I consider equivalent classes of maps from a source core curve C with n markings I Want the curve to be rational and I want The map to be degree D so push forward of fundamental cycle of C to be equal to D and I consider this up to automorphisms of the domain map and The bar over here, which I'm going to extend over here as well Is that I'm considering a compactification of the space where this map where this curve is smooth by allowing ordinary double points on the domain curve and I will request that the map is only allowed to contract components when they have a finite automorphism group now This modulate space is It has many downsides in general it's often not nice in the sense of lotters thought before But this compactification does at least the job of giving you something compact. So it's a proper Dilling-Monfer stack With an expected dimension equal to the dimension of X minus 3 Plus the anti canonical degree of the map plus the number of punctures and In the smoothness case, this is often as close as possible as being unobstructed. So it is actually nice in In the cases that we will consider but regardless it supports in general a virtual fundamental class living in pure homological degree equals to twice this expected dimension and my thought for counting nxd So this is nxd of say B nxd of B is given by an integral against this virtual fundamental class pullbacks of Poincare duals to my cycles by evaluation morphisms Yes, no, no at some point There's there's a word refined in brackets that you should understand as how you're genius in some sense And not in other senses of refinance. No everything is unit zero to this point Did I put the genius somewhere by mistake? I think I've been consistent Okay, so this knew nothing about the divisor D and that's what I'm going to introduce now so the first enumerative theory that I'm going to touch to a Smith-Nef pair xd is the local again g to zero Drama with the theory what I do with D is I construct a Vector bundle over x which is given by the sum of the Dual line bundles to each component of D xd local is the definition the total space of Our sum on to R the example one of p1 minus 2 points I'm considering the total sum of o minus 1 plus o minus 1 on p1. That's the result corner fold example number 2 was D this with cubic so I have o minus 3 and p2 there are two examples where o minus 1 plus o minus 2 and o minus 1 plus o minus 1 plus o minus 1 So this is a Kalbiya m plus our fault because D is anti-canonical and As I did before I would like to compute numbers again informally count Degree D curves rational curves in this total space when well a stable map To this total space is the same thing as a stable map to to the zero section at least when the intersection with The boundary divisors are Generic what reason is that since each of these guys Neff My stable maps are not allowed to deform holomorphically of the zero section There's there are just no sections before doing that So it's a scheme this modular space is exacted modular space that we have without the bundle however the the two obstruction theories differ and And This is codified by the fact that there's a canonical obstruction bundle on the modular space of stable maps to x the base His fiber at a smooth model I point like Curve C map phi and marking speed is the H1 pullback by phi of this this vector bundle and The virtual fundamental class in this case is defined as the intersection of the virtual fundamental class for the theory of the base with the top turn class of this obstruction local m0 and bar xd were intersected with the top turn class xd and Just by simple counting off the number of will dimensional this vector space just by certain quality you're cutting down the expected dimension by the anti canonical degree minus the rank of this vector dimension is Is m plus r minus 3 there's a second thing that I can do with my divisor and It's imposed sort of boundary conditions to my map and consider the relative theory of X to this divisor in this case informally I'm interested in a different set of numbers There's a b over here, and there's a through b over here. I want to count the numbers of degree d rational curves in X Through b with the condition that they intersect each component of the boundary divisor in a unique point of Maximum content order now there was no issue with properness in the ordinary case and There was a for theory no issue with properness in this local case It's every all the action was happening in the theory of the base But the moment you consider relative stable maps you'll run in the problem of having a compact modulized space So if you if you have a relative stable map with some prescribed Order of tangency at a divisor you have a sequence of those You can have a limiting stable map where some of the components of your curve fall entirely into the divisor so that in the limit the notion of Specifying a tangency doesn't make any sense and so this has been a source of Major headaches both in the algebraic and symplectic category and one And possibly the only one in the case in which we have more components Way to deal with it is to view X Through the lenses of log geometry So you will be with you X as a log scheme With a divisorial log structure induced by D Now somehow this means that informally you come to terms with the fact that you can have some components falling into the divisor But you remember the fact that you have some some contact order with this divisor by some homomorphism of ghost sheeps in the setting of log geometry If you're unfamiliar with this, I'm just gonna list the upshots of dealing with this modular Compactification, so there's a modular space space of log stable maps xd degree D With plenty of nice properties well first of all this is proper and Of expected dimension the same as that of the local theory Under some some canonical lift of a notional stable map to the log world and If X is log smooth, which is the case for us it supports a Natural perfect obstruction theory giving you a virtual fundamental class of the expected dimension And drama with an invariance relative drama with an invariance can be the form in terms of by integration against this Again as was done for what the local and the global theory Okay, so let me wrap up since I'm like halfway through my talk I got this fair given by a variety and a divisor in it and Starting from them from that I do two very different things in one case the divisor defines an embedding into higher dimensional Non-compact Calabi all manifolds and I look at the grama with them theory of this guy with no strings attached and The other thing that I do with my divisor is that I impose in quotation marks boundary conditions So I just impose contact orders at each of the components of this boundary divisor And I'm just looking at the setting in which this This tangency condition is maximum just a unique point of maximal contact order for each device I Priority there's nothing whatsoever that These two halves of the blackboard have to do with each other Other than depending both by the same ingredients and having the same expected dimension so somewhat surprisingly it was proposed towards the Well less than two years ago That there is indeed the universal relation Between the local and the relative grama with them counts. So conjecture log local principle of the title of the talk and this is due to vanguarel Graber and root dot is that there's a universal relation these two Sets of invariance differ by universal factor. The relative invariance are multiple multiples by a factor given by Minus one to the intersection of D With my jth component of the divisor times d times dj Times the local in there Before I give arguments pro and against and you will only hear the pro side because there's no against until now About whether this is true or not one thing to to notice is that there's a This is an identity between something relatively new this login variants That I was talking about that have a tendency to be closer to being a numeracy than ordinary grama with an invariance With something that's pretty much old at least to well studied for more than two decades And for which we know plenty of information from the calculational point of view via local mirror symmetry So there's a lot of power that we have over here that we can transfer to Do actual calculations that are pretty scarce on the left-hand side If this were true, then is it true well in the in the case in which the divisor is smooth This was proved by vanguarel Graber root dot in the original paper and And Then two papers appeared on on the archive last month on the general case and and they're pretty much orthogonal in the methods So first of all, I'm very happy to Give a plug to a beautiful work of nabijou and Ranganathan in August this year where They proved the conjecture with a mild positivity assumption on on the eyes So the de eyes have to be hyperplaced actions and that's essentially by Clever read reducing to the R equals one case and then use that the original FGGR theorem and So that that's about establishing the equality and in an orthogonal direction. There's a Rather more combinatorial result Which is what we obtained with pyrrhic busso myself and Where we don't prove the quality we just solve the full gram-width in theory engine is zero for the left and the right-hand side in a variety of cases There's there's enough. I mean, are you asking whether there's a log? I'm not up to speed entirely with the literature. So you're asking whether there's a log version of the Graber ponderic on the virtual localization Without any extra effort to be put and actually don't know it No, I don't know. I don't know. So what we do is Probably need slides to to present the statement of the theory It's going to be a little colloquial, but basically the result is that we can solve both the log and the local Grammar-width in theory is in June 0 for xd either Toric, so we actually compute both sides of the correspondence and solve the theory completely and Once you solve the theory can ask yourself is the principle true and The answer is yes So start start holds for the resulting invariance Now the way this we have the complete solution of both sides of the of the Correspondence is on the left-hand side, there's a tropical correspondence theory, which is a general thing in in log-gram-width in theory or in log geometry and This log-gram-width in counts are reduced to tropical curve counts with some multiplicity And what's special about these smooth and F pairs is that the tropical multiplicity can be determined in close form So I should probably add this in close form. So we have close form formulas for these invariants as functions of a degree and say the weights in the g it presentation of these guys and On the local side, there are equilibrium theorems of codes and given tile that we can adapt to our case and we can extend to To the full B quantum comology using some vanishing properties of the worksheet corrections to the mirror maps for for these special targets so there's just this Random observation that things are This smooth and F condition is enough to make things calculable on both sides. So a This is Probably bound to work In larger generality and B what we do prove is that we can drop the smoothness Condition on X or D. So probably pretty much everything I said works for variety replaced by Dling month for stack. This is something that we prove in the Toric case so this is bound to to work more generally and Speaking of Generalizations, there's since it was a request in this case All this story is lives By construction in junior zero. Is there any hope of extending this to the to the higher genus theory? Now, this is basically some It started off with some sort of annoying We're exciting and annoying result and I'm going to restrict my Discussion the case in which XD is a locking up here with two components of the boundary Now the the expected dimension in this case is What's M plus R? Minus 3 plus N So M is 2 and R is 2. That's an M plus 1 for the junior zero theory and for the higher genus theory This grows linearly and we have a plus G That enters in the expected dimension so if you want to define some Dimension zero virtual count what one typically does in grommet Whitton theory is that you add extra Incidents conditions so that you reduce to something that that gives you a number So that's something that you can do But I will want to do something else so for example if I start with one mark point and in junior zero I Have a two-dimensional moduli space so I can offset that by an insertion of a punk or a dual of a point and in juniors G Normally, I would have to add extra insertions, but instead of doing that I Consider something different which is I define to be as the integral over the virtual fundamental class in juniors G with one mark point of X and D and I retain my insertion of the point and I need something that's Degree G to get something that matches up with the dimension of the virtual fundamental class And my choice will be to use minus G times lambda G So the top-turn class of the hodge bundle This mom for its So I've got a vector bundle on on this moduli space whose fiber at a smooth moduli point is the space of holomorphic differentials on my curve That's a rank G vector bundle and I think it's it's top-turn class that I intersect with that now So Has a remarkable Prediction, which is actually a theorem About what this genus gene variants should look like in terms of the gene zero one or an appropriate modification of the gene zero one and The statement is that this is the log to G minus one to G minus two Q Taylor coefficient at Q equals one so at log Q equals zero of Some explicit Explicitly computable Rational function of Q sorry Q Q plus minus one and the way you move to from the genus zero invariance to the all genus invariance and in fact to this all generating function is by taking the tropical count that computes for you the genus zero invariance and Replacing the tropical multiplicity with its Q n log. So there's actually a natural tower. There's a actual Natural generating functions for the higher genus theory that live wholly on the locals on the log side and in view of the log local principle You are compelled to ask yourself whether there's any local interpretation for this relative higher genus relative invariance and This has put some pressure on the on the local side of this collaboration aka yours truly here because there is it's it's hard to come with a natural definition of What this refined higher genus local invariance? what they should be and One problem with that is that if you go to higher genus Instead of increasing the dimension your expected dimension of the modular space as you did for the case of the surface Your expected dimension here becomes negative As soon as genus is greater than one so there are no invariance to play with in in the first place So the natural guess I will take higher genus local invariance that just doesn't work because there's nothing There in the first place I'm not on the Calabi out free for because I'm on a surface and I'm at I have our equals to so I'm on a Calabi out fourfold That's exactly So in a Calabi out fourfold you have either point insertions on the surface in genus zero or Unpointed invariance in genus one and then nothing in higher genus So with well the answer to this is we've come up with three refinements They're all supposed to coincide One dates from Thursday night, so I'm I will not be so brazen as to just give a talk about that today One is a couple of weeks old, so that's not very mature, but I will give her a mark at the end about that and So let me give you the more mature one Which is what we call Summiting properly The refined Well, the conjecture is as follows If you take a locking up air With two components for the boundary divisor you can attach to this fourfold a pair Yd D Yd is a Toric Calabi out threefold and LD is a Toric special Lagrangian brain inside Yd such that the following holds First of all, they are genus zero local invariance What insertion of a point are equal to the Generating function of holomorphic discs in Yd with boundary on this Lagrangian so we have its Dimension zero virtual counting problem here and in this Calabi out three setting even when we consider the open sector This remains genus zero for for all general. So in particular we can ask your ask ourselves What if we construct generating functions window of Q Such that These are defined To be the genus G open-gramma weighting counts in Yd LD and The second condition So the first needs to hold and the second is that the relative invariance of the pair xd in D are Given by We have the same sign factor that we had in the unrefined case and The intersection multiplicity is that we have in the unrefined correspondence get cutiform. So I take the Q analog of this number and I multiply With my refined local invariance I Guess I should I have 15 minutes. I'm going to use five to motivate this I Have no geometric intuition why this has to be true But there's some well-defined thought process that leads to postulate this sort of correspondence and reason is Well in one word is four-dimensional Calabi of four geometric engineering Plus some generalized open-closed string duality Which is due to Petter Meyer Almost 20 years ago. So one way to one time on her way of Try to dig out this sort of statements and in the context of mirror symmetry is to See what these things compute in real life for example in some target space interpretation of The topological string theory and see how you can relay them through some some chain of physical dualities Now if you want to ask yourself if I take a calabi out for and I take a type 2a Compactification on a calabi out for well if I did it with a calabi out 3 we know that the For example gauge theory pre-potentials can be computed by geometric engineering on just as an effective theory coming down from this Calabi out 3 compactification So if we try and play the same game in by compactifying down from 10 to 2 dimension We get a holomorphic super potential term Computed on on an effective and equals to two theory Attached to this calabi out well the fact that you're compactifying on something non-compact just means that you have some non-gravitational theory you're essentially pushing the the Planck mass of infinity and reduced to some Honest and equals to two gauge theory But there's not the only way where you can compute topological couplings in in a string compactification You can consider Wrapping d4 brains on special Lagrangian And you look at what happens on on the transverse direction, which is two-dimensional And again the discample you compute exactly the same type of term in the effective action But more over there's a there's a whole tower of couplings With the gravitational chiral super field, which you don't see in this four-dimensional perspective Which is why we don't know how to give a Just just purely in terms of the geometry of this calabi or for an interpretation for these invariants But we can't do it through through open grommet within theory on on this special Lagrangians and Well a fact is that to any to any XD log We can indeed associate a pair yd LD by partially inverting the my recipe for going the other way round and Once you have that Last five minutes may or may not have made the blindest bit of a sense to you. That's fine What I have is that for some irrational reasons or not I postulate that there's an identity between this refined local invariance computer in terms of open grommet width and counts and their relative invariance with insertions of higher genus Lambda classes It's a statement. Is it true or not? Well, you can prove it or disprove it and one Piece of evidence towards the fact that it is true is that it is true we can actually check that all this higher genus open curve counting in The calabi o3 pair with special Lagrangians attached to a the local space from a locking up pair Match with this prescription to with with a login variance The two other refinements I don't have enough time and it's it's too young a subject For me to actually tell you something sensible about it Let me just conclude about so one When I wrote down the unrefined log local principle, I Was saying that the relative invariance are something for which we don't have any almost any explicit calculation so far and We can the use of the principle is that we can transfer all our local mirror symmetry wisdom from 1999 onwards over to the left-hand side and The obvious question is whether we can do the same at this refined higher genus level. Do we know more? Or do we know less about this higher genus setting? in this restricted case of two two component locking up pairs and Well, the answer is quite I'm just gonna keep it brief there's There's a lot of structure that has been attached to the local theory of Toric Calabiel three-folds with the grunges in them after the work of Well, I'll breached Marko's and and collaborators in in the last 15 years in particular whatever applies to the generating function of open ground weight tuning variance for this Toric three-folds can be transported over by this But the refined local principle log local principle to the log side. So in particular To any locking up pair we can attach a matrix model a crystal model a tile function of some integrable hierarchy, which is a reduction of the two-toda hierarchy and family of spectral curves such that the Modeling the b-model Conjecture slash here and now after work of Melissa Liu Fong and Jew Computes the log grung with an invariance with this lambda insert is at higher genus. It's just something that cannot have been Presumably predicted without any sort of relation with this parent theory Another thing The refined principle accomplishes is that They're a strict Integrality Statements that hold true for this open grung weighting counts in terms of open gopacum or Vaffin variance Well an open version of gopacum or Vaffin variance So it was conjectured by by Pierre egg in That's right. Yeah It was conjectured by Pierre egg that generating function of log grung with an invariance in a different setting when you have only one contact order at naturally the form of Oguri Vaffa partition functions, so the log grung with an invariance were naturally Open bps invariance for some putative theory of count curve counting theory of three-folds That it was difficult to visualize What the principle does is well It gives you that theory and it gives you the identification with this actual open bps invariance And this also holds true for the closed theory of the fourfold itself So in particular if we can establish mathematically an integrality statement about these open bps invariance We can prove Integrality statements at a level of this local theory for the fourfold and this was this is a conjecture of Albrecht with Rahul Pundari Pande in 2008 where they proposed some definition in terms of multi covering formulas of closed bps invariance for this for fourfolds and Whatever we can establish at the level of the threefold with a brain we can transport over to the fourfold theory and it actually turns out That we actually can because a further piece of free information that we get from the from the log local principle is that there's naturally a quiver theory and An analog of the not squivers correspondence This follows by work of Kucharski Reineke and Sulkowski Such that the unrefined DT invariance coincide with the unrefined log invariance and in particular with the closed Gopakumar Vafa putative integral invariance in the middle so we can actually prove this conjecture for fourfolds and I like to use some of the time to Tell you something about so this in particular this this relation with DT theory Comes with for free with a with a further refinement by consider the motivic story on the DT side and We have very encouraging signals that this actually matches the open ground with the story that I presented before And yeah, I guess I could stop here. Thank you