 Thanks for the invitation and thanks for this opportunity to introduce this work at our conference Before we start the actual talk it might be helpful to just briefly explain the motivation of this work So we all know that there are many different ways to define the Co-homology of a finite dimensional closed oriented manifold and one possible approach to use a Morse function on this manifold It was Witten's original idea that if we only want to define this Morse co-homology It is a fascist to just count critical points and suitable negative gradient flow lines of this Morse function Then for realized that this particular approach can be generalized to suitable infinite dimensional cases Producing powerful invariance in both implanted topology a low-dimensional topology and our question here is how do a complex by this story? So what if we have a killer manifold and a holomorphic Morse function? Then what kind of algebraic invariance were supposed to define for such a pair? This pair is usually called Landau-Ginsberg models in the literature As we will see in just a moment if we just consider the Morse co-homology of a holomorphic Morse function Then it would be rather boring because essentially you are just counting the number of critical points But thanks to the work of Paul in the finite dimensional case We actually have very good answer to that question so we should really define the Foucault-Seidel category of a Landau-Ginsberg model and in our context this just mean Directed the infinite category with finitely many objects and The end point of this project is to generalize a well-known Result is implanted topology to help us understand the cyber-weight-engaged theory So this project is not fully accomplished yet, but it still seems pretty Promising so let me start with the first part of the talk and explain the expectation from the finite dimensional case So let me start with the definition an LG model is simply a pair where M is a non-compact complete killer manifold and W is a holomorphic Morse function Defined on M. It's also called the super potential in the literature So I will use L to denote the real part and edge for the imaginary part and here the Morse condition Just saying that the real part of the super potential is the Morse function is ordinary sense So it doesn't matter Whether you use the real or the imaginary part because the Kirchhoff-Reyman equation says they have the same critical side At this point, we need some extra condition for this Landau-Ginzman model for instance I want this killer manifold to have bounded geometry and Also one this implicated manifold to be exact this means this simply two form is the differential of some smooth one form called primitive Well, I also want this function The norm of this gradient vector field now the edge to be a proper function on the killer manifold This implies in particular that there are only finally many critical points for this holomorphic Morse function In practice, we have to be a little bit more precise about the growth rate of this function at infinity It cannot just grow too slow or too fast But I don't plan to elaborate on this. Let me just say there are some other assumptions here They help us control on the geometry of this Landau-Ginzman model and infinity It's more enlightening to just have a few examples in mind So for the first one, let's take M to be CN and the superpotential is a sum of the I squared Then the origin is a unique critical point as the norm degenerate and for the next one let's take M to be C and W is just a polynomial and Then Morse condition in this case just says The first derivative of the superpotential Has only simple zeros and the first example is always the local model near a critical point of this holomorphic Morse function and let me give you two more infinite dimensional examples So for the first one, let's take Y to be any closed oriented three manifold then and this is the space of SL2C connections So this is infinite dimensional offline complex subspace So this means so any element in this scalar manifold takes the form on the trivial differential plus some smooth one form and B takes value in the algebra of SL2C So the topology topology of this space is rather simple The superpotential in this case is a so-called the complex transformers functional So the next example is more relevant with our several written equations So let's take a sigma to be a closed remand surface of some genus G and Let's also choose a line bundle over the surface of some degree D then in this case M is still modeled on an infinite dimensional complex vector space So an element in M is really a triple where B is a U1 connection on L and Fee plus is a smooth section of this line bundle and Fee minus is a smooth zero one form Taking value in this line bundle L The superpotential in this case is called a direct functional and its definition is rather simple So given any triple like this You First apply the D bar operator of B to fee plus then you take the Hermitian inner product with fee minus Then you take the integral over the whole surface Then you multiply or normalizing constant So this functional on the nose is not really a Morse function So you have to add a super perturbation to make it Morse and We usually do this by choosing a harmonic one form on the surface So there is a caveat when working with gauge theory because there is always an infinite dimensional legal practice on the killer manifold So we are secretly working with so called the gauged algae models But the generalization to go from the ordinary version to the gauge version is not so significant So I won't really emphasize the distinction in our talk today So returning to the finite dimensional case We should really think of this super potential as a projection map onto C. So The fiber over a generic point is really a smooth complex a manifold of M But there are a few singular fibers One for each critical point of the super potential Within this killer manifold there exists a distinguished Class of Lagrangian some manifolds called symbols. They are simply defined as the unstable or stable some manifold of the real part of W For each critical point on there is a pair of stable and unstable symbols. So this is the right So this red one is a picture for the stable Thinball and its projection onto the complex plane is always a ray emanating from this critical value and the reason is the function H the imaginary part is always constant as such a thin ball and if we work instead with The unstable one The projection would be a ray pointing towards the opposite direction and from now I will only draw the picture downstairs on C So I will use a Critical value to secretly represent the critical point upstairs and for each of them on there is a pair of Stable and unstable symbols Thinballs are very important They are Lagrangian some manifolds and they generate the focale category of this reference vibration in a suitable sense This implies in particular that the Morse index of each critical point is Always half of the dimension of the total space And so here with let's make one extra condition here So let's assume the value of H are all distinct for this critical points And this condition further implies that there are basically no flow lines going between two critical points So the Morse homology of this real part is just counting the number of critical points So if you go back to this example a critical point of this complex transsemmons functional is actually a flight as L2C connection So it defines a pi y representation of the fun of the three manifold into as L2C and Essentially, you are just counting on Taiwan representations So this is relevant with the work of upside that my last school a shift theoretical model of SL2C for a co-homology So the problem is that it's not guaranteed that this complex transsemmons functional is always Morse So you have to introduce some extra techniques to make a reasonable count For this the last example here The critical points of this direct functional do not really have a very good geometric meaning But you can compute the rather concretely that up to gauge this critical side has precisely 2g minus 2 choose D Critical orbits so you can make everything very explicit for this direct functional So returning to the finite dimensional case once again. So in 2008 I'll prove the following theorem using his deep generating theorem for the focacate great in his book. So for any Compact exact Lagrangian some manifold in M. So here exact means When we restrict the primitive one form on X and on Y They are going to be some exact forms Then there exists a spectral sequence converging to Lagrangian focal homology between X and Y and whose UN page is given by this direct sum So and each summand is given by the focal homology of X with sj Tensored with the focal homology between uj with Y and This summation is over all critical points of the super potential You should really think of this result as a computational tool to help us Compute the focal homology between X and Y and the moral is if we understand how it how this The first Lagrangian acts interact with all stable symbols and how the second Lagrangian Intersect with our unstable symbols Then we can compute each page of this spectral sequence step by step and eventually compute this infinite page And the upshot is only finite finite amount of information for X and Y is required in this computation Okay, and our goal is to generalize this result to suitable infinite dimensional cases So this complex transformers function are extremely difficult to work with but this example might be promising So we have to read Do it the right way to make this generalization possible. So Paul's original approach to this theorem Realize on his generating theorem. So is only constructed algebraically So I will call this the algebraic spectral sequence for the rest of the talk Also in case proof you use the some notion like a simply parallel Transipation and the simply then twist along Lagrangian sphere and and I don't really know how to generalize those notions to the infinite dimensional case so for the rest of the talk I would like to Describe our turn here approach to this spectral sequence and in particular I wish to see the filtration on this coaching complex rather explicitly and to this end I really have to explain the construction of this for co-homology groups Before I go any further either any questions so far about this setup Great, so you have to choose the complex structure very carefully to make this guy holomorphic So the complex structure J would be the for the first entry is how to start the second entry I This last one minus I very good question. So this is the second part this talk So let's start with the last group the coaching complex between X and Y Traditionally this is defined by counting on J holomorphic strips So there are simply Smooth maps from this infinite strip into the killer manifold and there is a jab to the cushy Rima equation And then we have Lagrangian boundary conditions along as Along well as is equal to zero and equal to R So this equation is conformally invariant So it doesn't really matter which last parameter you choose here But in order to see the filtration concretely, I really have to perturb this equation using the super potential So at this point I will use the gradient lecture field of H And then this equation is not Conformally invariant anymore. So the choice of R becomes important So let me add a subscript here to indicate the dependence. So I will call this equation the foreign equation in this talk So to some extent on this perturbation seems to be a more natural equation to consider for a long dog is one models for instance For the complex transimons Functional on this floor equation will recover the so-called Haley's written equations on this product find manifold Z cross Y and And for the Dirac functional This floor equation will recover our favorite several weighting equations This product for manifold Z cross sigma as I mentioned earlier if you want to work with gauge theory There are some Infant dimensionally groups acting on this basis and we have completely ignored this action in our Discussion so far. So this reduction or specialization are not so accurate But it's harmless to think of this floor equation As a finite a toy model for this gauge through radical equations and this particular perturbation will produce the right filtration for us So this is the construction for the last group. So what about the other two another difficulty in generalizing This results to the infinite dimensional case. It's really about the Lagrangian boundary condition Infant dimensional Lagrangian some manifolds are extremely annoying to work with so I would like to get rid of the boundary condition whenever possible So in this case X and Y are just some general Lagrangian some manifold So I cannot say too much about them, but let's say for the first group of there Which is between X and SJ. So the second entry is a Thin ball so we can propose an alternative rule for this coaching complex. So this time I'm going to study this Floor equation on the upper half plan We still have a Lagrangian boundary condition along the real line But the other part is replaced by asymptotic condition at infinity So I want this solution to converge to our critical point qj if we go up vertically upwards So if I let S goes to positive infinity this would be our critical point qj and this Convergence would be exponential and uniform in the time variable at this point, you may wonder why This upper half-pan model is producing the same for a cohomology group for us So let me try to provide some heuristics here. So to simplify the situation. Let's suppose That the time derivative of u is that then call it zero So we side this derivative to zero in the equation then The question remain question says lambda h is also j of lambda l So this floor equation implies that the spatial derivative of u plus lambda l is equal to zero This means if you go upwards here, you are following the naked gradient flow of the real part So let's give them a name. So they are Called solitons Because there are solutions independent of the time and we use c of x comma sj To denote the space of all solitons So it follows the naked gradient flow of l the initial point less on x and the limit as As goes to infinity is the critical point qj So if you think very carefully about this initial point this corresponds to a genuine intersection of This Lagrangian some manifold x with a stable symbol So this is still aligned with the traditional Lagrangian intersection theory and In a similar way let's define the other group by counting solutions on the lower half plan So we still have a boundary condition around the real line But the other guy is replaced by asymptotic condition as s goes to negative infinity So we are always using the same floor equation and we again use c of uj comma y to denote the space of solitons And if you think carefully about this ending point this corresponds to genuine intersection of y with the unstable symbol and finally We use c sub r to denote the space of solitons or The first group at this point you may wonder why am I changing the setup like this? You see this immediately some magic is going to happen So there are two basic observations at this point So for any R suffinger large There is a very nice Decomposition for this space. So c sub r can be decomposed as this disjoint union and Each salmon is in bijection with c of x comma sj Cross c of uj cross y So yes Position of y is in there. So it's always along the other boundary component Except why are fixed? I will take R to be large instead. I'm stretching the neck it Well, you mentioned Alternatively, you can fix the domain and you put some R here, right? So if you take R to be zero, it just recovers the ordinary Christian equation You take this one to be one you take this you multiply a big R over here They're just rescaling the domain X and Y can intersect so So This bijection follows from the standard gluing theorem from in more sweeten theory So a flow line that goes from X to Y Will eventually degenerate into a broken flower line that passes through some critical point QJ and our assumption here in plus that only one break is possible and You can choose which critical point this flow line is converging to in the middle of this large interval So you see this decomposition and this bijection is simply given by the gluing theorem So the first the second observation is equally simple, but it's that it's more Substantial so by the first step we can decompose the coaching complex between X and Y As this direct sound And a VJ is just freely generated by the summon. So here comes the question. So this is the Coaching group. What about a differential? To understand the structure of differential So let's take a look with the action functional. So you can think of this for co-homology as a Morse co-homology as an infinite dimensional space and the functional is the standard Simplactic action functional And this is the only payload place we need to see this formula So this formula takes the following form So they are basically four different terms So at the beginning we see the value of hx and h sub y the third one is the integral of the primitive one form and the last one is The integral of the value of H Okay, so here is my question suppose This soliton lysine the j-summon was the approximate value of this action functional I think all for you can figure this out in just 10 minutes, but I will spoil the question anyway, so suppose Gamma lysine the j-summon then the answer is this action from functional is equal to r times H of qj Plus big of one the reason is also straightforward because as our ghost infinity this Blow line gamma will converge to a broken trajectory like this. This means that the first three pieces Will actually converge as our ghost infinity But the last term will actually blow up and the reason is this path Gamma is almost a constant path and This critical point qj. This means this function h is almost constant at the value h of qj And when you integrate over this large interval, you'll see this linear order term and all remaining terms are going to be bounded so this property is very important and because The floor differential can only increase the value of the action functional. This means if r is sufficiently large The floor differential can only increase the value of edge So let's come back to this picture. So for each of these critical points. We have a salmon We want the two and we three And there might be some floor differential preserving each salmon. That's completely possible and there might be something that Increase the value of edge, but there can't be anything going backwards This means under this decomposition. This is a lower triangular matrix inducing a spectral sequence So here comes to the main result of this talk so of for any r is sufficiently large This energy filtration induces a geometric spectral sequence it starts with the cohomology of this salmon and converging to the four cohomology between x and y Moreover, you can compute the first page of the spectral sequence rather explicitly So the cohomology of your salmon is Isomorphic to this tensor product So we have already seen such an isomorphism between their chain groups and the claim is that you can also identify their Complexes so you can identify their differential maps under this next stretching limit So the second statement is this geometric spectral sequence is esomorphic to pose algebraic spectral sequence So if we just want a spectral sequence, then part one is already good It implies a ranking equality between the e1 and infinity page But the problem is that this filtration geometric filtration itself didn't say anything about the structure about hair Differentials suppose we want an algebraic prescription to really understand the differential on each page and Eventually compute this for a cohomology group then this algebraic spectral sequence becomes more fundamental And in order to pursue some result like this We really have to define the focacil category of a long noggin's remodel within this framework So before I move to the next part, are there any questions so far about this statement and especially about the origin of the filtration This is really the main takeaway from the talk By the symplactic action functional, right? So the four cohomology is the most cohomology of the action functional, right? And I can estimate the approximate value of this action functional for each assignment So the leading order term is going to be very important. Okay. I think I'm running out of time So I will just give you the statement of The result so in our case the focacil category of this long noggin's remodel just mean Directed infinity category with finitely on Finitely many objects and these objects are given just by these all stable symbols They are ordered to increase the value of edge and this infinite categories directed means that the home space between sj and sk is 0 if j is greater than k and is Generated by the strict unit if j is equal to k and it's given by the coaching complex When j is less than k. So only something interesting happens in the last case At this point you see another problem. So these symbols don't intersect at all in the killer manifold, right? So what's the plan? We have to do some very interesting perturbation to this picture and I would like to rotate each symbol anti-clockwise by a very small amount So this means so this is the perturbed picture Let's fix a base point that is far away in the real direction And I would like to ask all perturbed in those to pass through this base point So this means we have to choose some small angles Thread 1, Thread 2, Thread 3 and Thread 4, one for each critical point And I would like to replace each stable symbol by sj prime and this is defined Let's first rotate by the super potential w by this unique complex number e to the minus i theta j then we take the real part and define sj prime to be The stable some manifold of this guy and then I just replace everything by this perturbed symbols And the consequence of this perturbation is that if you think about the projection of each stable symbols this time They are now pointing towards the direction of e i theta j and we do the same thing for all unstable symbols at this point you can just work with Lagrangian boundary condition to define these groups But since they are all symbols there is a way to really get rid of Lagrangian boundary conditions And by and this is done by considering some interesting perturbation of the previous floor equation R2 But let me skip that part in this talk There are a few more ingredients In this actual spectral sequence, so let's Also define the possible do category of Of this algae model, so this is defined all using all Unstable symbols so the rows between stable and unstable symbols are pretty symmetric at this point The next one is more interesting. We want a diagonal bi module For a and b so So this by module is defined by putting all stable and all Stable symbols together and by definition for a by module. I have to assign a chain complex between each pair UK and SJ In this case, this is just the co-chain complex This guy is called the diagonal bi module because this Complex has a particularly simple structure so it is zero if J is not equal to K and Is one-dimensional When J is equal to K and the reason is rather simple So if you choose different indices then the symbols stable and unstable symbols They don't intersect at all in a catered manifold But if you choose J equals to K then the intersect at the unique critical point QJ and So the stable so The first set of category a and b they behave pretty much like the basis of vector space and the dual basis of the dual vector space and this diagonal bi module somehow Manifest the duality quality duality pattern between a and b basically one determines the other and the final leaf for x I want to associate a Infinity left module over a and for the second Lagrangian I want right B module and this point We are ready to stay the refinement of the theorem So this is really a chain-level green theorem. So for any R is sufficiently large There exists a quasi-isomorphism that goes from this coaching complex between x and y To this huge complex build out of the infinite data so here P is the DJ category of of Left a module and Q is the DJ category of Right B module More on the left hand side that we have the geometric filtration induced by the action functional and on the right hand side, we have the algebraic spectral filtration Induced from this right B module. So for any right B module There is a canonical way to filter this right B module by a sequence of some modules and this induces the Algebraic filtration on the right hand side and this quasi-isomorphism induced identifies these two filtration Finally for this result, you can simply ask x and y to be compact This is completely fine, but we can also be a slightly more general So we can ask instead that the real part of the super potential x is bounded from above and On y is bounded from below So in this picture So the only thing we know about y is the projection is bounded from below in the real direction and for x We only know it's bounded from above and this is the reason I have been using different Modules for x and y because they are not symmetric at this point. So In the last minute, let me just explain briefly the dictionary between Simplactic topology and several we can gauge theory so for the symplactic action functional a configuration is Pass that goes from x to y and in several we can gauge theory So this x is replaced by a three manifold with the boundary y1 and this interval is replaced by an interval Cross this boundary remand surface So it could be disconnected and y is replaced up by another three manifold with boundary and The action functional is then replaced by chain samples direct functional and the floor equation Is replaced by the cyber written equation on this three manifold across r so when you change on the length of this Interval so making it longer and longer You are secretly stretching a neck for this three manifold You are changing the metric and you will also see a similar energy filtration for the transom is direct functional if you well have a Very interesting perturbation in the middle So the geometric spectral sequence Is always easy so we always have this spectral sequence and the real challenge is to generalize algebraic spectrum framework to This situation so this party is not fully accomplished yet, but it seems pretty promising at this point So at the very end of talk I have to say that I'm very fortunate to benefit both from the work of Tom and Paul When I was a student at MIT, I remember that in my third year, I came to Tom's office and said you know to understand the cyber written equations on the complex plan cross a compact remand surface on the correct finite dimensional model is To start with a killer manifold with a holomorphic Morse function and Tom started to explain to me that this is called a long dog is one model and I started to learn Paul's work on that freeze vibrations and can't grace and I think it's quite remarkable to see such a connection Between their work and I hope I can tell you a more complete story in the near future Thank you so much for your attention one second again, so That the point is We have a decomposition so for The coaching complex between x and y is freely generated by the space of solitons and we can have a decomposition Like this and this is because each flow line will eventually converge to a broken trajectory that goes from So it's initially goes from x then to some critical point then into the second Lagrangian and the upshot is so This at this point you really have to see the formula. So the action functional For a gamma in the j-summon is approximately R times h of qj So this is a very simple but very important observation then The differential can only increase the value of edge Yes well So this formula is already suggesting Such a border multiple fluid theory, but it's not going to be a very general result So imagine that we can define for the direct functional from for some a infinity Categories a and b and for y1 y2 you can define this left and right modules And you can you also have this diagonal by module in the middle. So this is already somehow the gluing formula for multiple for homology, but the point is The direct functional is not really a lack super potential onto C So it's really a leftist vibration over two torus. So in some sense we have to develop a Picard leftist Novikov theory and And so anyway, so he's going to be a little bit more complicated than this