BPS Counting and Pseudoperiodic Topology
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Kontsevich, Maxim
A holomorphic quadratic differential on a complex curve defines a flat metric with conical singularities. In the case of simple zeroes, T. Bridgeland and I. Smith identified geodesic intervals connecting zeroes, as well as maximal geodesic cylinders, with stable objects in certain 3-dimensional Calabi-Yau category. As a corollary, the counting of such geodesics gives a wall-crossing structure in the Lie algebra of Hamiltonian vector fields on a symplectic algebraic torus. I will explain that essentially the same numbers give wall-crossing structure in a different graded Lie algebra, of matrix-valued functions on an algebraic torus (joint work with Y. Soibelman). This WCS makes sense for curves endowed with abelian differentials with zeroes of arbitrary order and can be generalized to closed holomorphic 1-forms on complex varieties of arbitrary dimension.
 Thank you very much for the invitation. So the topic of my talk will be slightly different from what you heard before. It will be kind of new direction in wall crossing. But some topic, in fact, it wasn't really addressed on this conference about wall crossing for three-dimensional Calabio categories in general. Yeah, so this story which we started with Jan quite a long time ago. So the idea is that you have triangulated category and some technical properties we call inconstructible objects form. Can be characterized by algebraic varieties, which is three-dimensional Calabio, and when we have stability structure, then it should produce certain DT invariance, certain integers, variable by some indices, and changing stability structure DT invariance by some certain wall crossing formulas. Yeah, so it's a whole story. But people in algebraic geometry, example of this, like the following three Calabio categories. Examples of categories from algebraic geometry are the following. You take three-dimensional complex manifold, algebraic variety, say, this canonical class trivial. This could be compact or not compact. And now you choose, pick some certain subset Y in X cube, it will be compact algebraic subset, and then one consider a category of perfect complexes on X with supported Y. Y could be a point curve, or could be Y itself, it's Y is compact. And in a reasonable situation, one can expect there will be some stability structures like Gizikir stability, and then we count roughly complex coherence sheets of stable pairs with support on Y, and get certain guys, and then we can recalculate. And I think Richard Thomas gave you some examples for such situation. We can get some Y as a surface, and X as a total space of canonical bundle on the surface. In algebraic geometry, it's pretty hard to construct the stability structures, and there is kind of mirror-dual picture. You consider some kind of six-dimensional, real symplectic, infinity-symplectic manifold, and flex-dual, because it's a real source of six-dimensional, maybe not compact one. And in fact, you also assume that it has complex structure, and canonical class trivial, and it has holomorphic reform, and objects which you consider will be, now consider something called Foucaille category of your variety. So roughly consider Lagrangian submanifers, but nobody knows exactly what is it. And then stable objects in this picture should be special Lagrangian submanifers. So it means it's Lagrangian submanifers, such as restriction of this 3-0 form has this constant argument, which I denote by theta. Yeah, here's a kind of different situation. We don't really know what is the category, but kind of know, up-rower, what are stable objects. Here we know the category, but stable object, it's hard to see. And these categories are not really constructed rigorously in any concrete case, but there are some things which we can guess what is going on. Namely, there was a story constructed by Tom Bridgeland and Ivan Smith. They constructed some category with stability structure looking on this picture. They don't directly identify it with Foucaille category, but obviously it comes from this picture. What they started with? They started with some non-compact algebraic curve and with quadratic differential. So this small cube will be section of curve and square of canonical bundle. The quadratic differential with simple zeros. And they have some numerical conditions, so the curve should have sufficiently many zeros and the form of quadratic differential should have full infinity. It's expected this should work for compact curve, but I don't think it's really constructed this category in this case. So what you do with this curve? Morally, you construct a, you do the following. If you consider cotangent bundle to the curve, which is a very complex dimensional variety, and it contained spectral curve given by quadratic differential. It will be plus minus square root of quadratic differential gives you a billion differential defined up to two valued section of these things. And it will be smooth curve, spectral curve to be kind of doubly ramified at zeros of your quadratic differential. And yeah, it's complex two dimensional variety and now I construct something like called conic bundle. So the bundle which have fiber C star outside spectral curve and C union C can be degenerate hyperbola over spectral curve. This will be three dimensional variety. And then this three dimensional variety one try to see what are special Lagrangians. Yeah, so you don't really think about what is scalar form, but one can guess what are special Lagrangians and the idea is that they can respond to one to one to the following things. It's called, what's called saddle connections. Make very, and close draw basics. What do I mean? If you have quadratic differential, then it get, take absolute value to get a flat metric. But with conic singularities because it's zero quadratic differential metric is not defined. It's conic singularities and conic singularities, the total angle will be three pi instead of two pi. Oh, sorry, no, sorry, conic singularities you get three pi, yeah. That's the angle. And so it gets a flat metric and then you can be interested in just kind of geodesic lines containing, connecting to zeros. How one can proceed, how this one can associate, this saddle connection can associate a sphere sitting in this three dimensional Calabio. How we do? If you got saddle connection, yeah, so this is your curve and here's a double cover spectral curve. First of all, what you'll construct, you construct a disk in a total space, namely for each point on the saddle connection, you consider two points of the spectral curve and connect by real interval. Because we get double cover of this thing, you get small interval connection, just two choices of square root, which degenerate to a point on the boundary, so you get a disk, you get two dimensional disk. This boundary, yeah, so from these things you get two dimensional disk in the cotangent bundle, as I did boundary, leaves in the spectral curve. Okay, now I have a bundle with fiber C star and the C star get middle circle. So I can see the bundle with fiber middle circle of interior of the disk, degenerate to a point and I get three dimensional sphere, oh, sorry, what, yeah, yeah, this way I get a three dimensional sphere. Okay, and one can convince yourself if you put appropriate very degenerate symplectic structures, it will be really special agrarian. And similar if you get closed geodesics, closed geodesics you get your S1 crosses 2. See you can see three dimensional manifold. Yeah, okay, so you got this hypothetical picture and what they really constructed, they constructed categories and these are objects and this will be simple stable objects. And construct stability condition, yeah, that's complicated work. And then by general think which is also very complicated to get full crossing formulas, but the composition of the thing is really elementary. So what here goes on, I get certain lattice which is H3 of this three kalabeya which constructed with integer coefficients. It has intersection form, skew symmetric form. And then I get a map from the slattest to complex numbers. Yeah, there are many ways to see it, for example, the slattest can be identified with H2 of cotangent bundle to my curve relative to spectral curve is integer coefficients. And then you integrate canonical one form PDQ on cotangent bundle. So get this functional and you get, and this whole crossing formulas, very abstract whole crossing formulas is coming from any three kalabeya categories to any lattice and the skew symmetric form. And DT invariance, they have contribution of two types, namely saddle connection, which gives me a sphere and then class or it gives you a certain class in this lattice and it gives, and this number will be one because it will be isolated sphere in just one. And for closed geodesics, for closed geodesics, then it's slightly different because closed geodesics, they automatically appear in one parameter family, which relate to the fact that it's one crosses two, it has first co-emulsion, it can move. So get one parameter family and, strictly speaking, what will cause on the following. You should consider not only Lagrange manifold, but rank one local systems, maybe this gives you non-recommendant field. So you get something like annulus in some non-recommendant field. Yeah, so what you form annulus of some non-recommendant field, but this cylinder has some kind of end points and at end points something complicated goes on the 4k category and this thing's kind of like compactified to project space to P1 and then it gives omega of annulus of closed geodesics will be plus two, the earlier characteristic of P1. Yeah, so it's some kind of explanation, but one can sit at the rest of the receipt, take number plus one and plus two for this guys and then if you start to change your quadratic differential on a curve, it means that you change a little bit stability structure on your things, these things should be recalculated in certain way. And here's some kind of basic identity, not dramatic, and this, so you get certain identity how the numbers are calculated and for this you don't need all the whole machine, you can do it elementary by hand, just drawing geodesics and gradient lines on a curve. And for example what happens typically, you have maybe three zeros and you get one geodesic and another geodesics, but here you get two angles alpha and beta and alpha plus beta is equal to three pi, but when you change complex structure in the form and then if alpha is equal to pi and beta is still very large to pi, then in this moment you'll have another geodesic connecting, if its angle is really less than pi, so you get a triangle and then you have another geodesic which appear. So you go from two geodesic, two saddle connections, goes to three saddle connections. So this is typical transformation what is going on and here's this wall crossing identity kind of famous in this thing which is, you've seen it many disguise, so you get identity, corresponding identity with certain elements in certain group which will be simplectomorphism in two variables, namely you can see that for any integers a and b, you can see the transformation just in two variables, variational transformation which will be clustered transformation x multiplied by one minus, you get transformation from variational transformation for plane to itself and then you get this identity, of course it's 0, 1 you get essentially just two choices for one, one will be one sign, one 0 will be different sign and all is based essentially on this identity and you also analyze something which happens with these cylinders and will be another infinite identity for such guys and how we can prove this wall crossing formula directly as going to this row to categories and this complicated machinery with whole story, so it's completely elementary. In fact it was known from many years there was something similar goes on for much simpler group, here the group is group called variational transformation of plane but now consider just three by three matrices, you can put three by three matrices and then there is similar identity namely one, one, one, one, one, all the rest is 0, you get this identity for three by three matrices and if you take a logarithm, get elements in some graded algebra, yeah it's for many years I was puzzled why there are such two completely different groups, they have inside this kind of bunch of elements, it's the same identity and the question, can one see this counting of subtle connections through this perspective, not through this wall crossing perspective and what the goal of my talk to explain is yes, there is different wall crossing formulas for counting almost the same subtle connection, I do not count closed geodesics now anymore, I consider count only subtle connections and there will be this wall crossing formalism which works for them, yeah so it's pretty surprising, you get two different groups and the products and the compositions are essentially the same, so and the second story, maybe I'll just tell you a few words what are all this wall crossing formulaic, what does a wall crossing formalism is in general, it starts with, it's almost nothing, so we have let's say a lattice of finite rank, like z to power n and suppose we have a Lie algebra, let's say overrational numbers which is gamma graded and so the commutator of the homogeneous element is homogeneous of degrees equal to the sum, just say Lie algebra and what I call stability structure on G is the following, is there is z and a collection of a gamma, z is a map from my lattice to complex numbers with additive map, so just enough to define on the basis of the lattice and a gamma is a collection of elements but for nonzero elements of the lattice, so it is saying only one stupid axon is called support property which is the following, there exists the quadratic form on gamma cross R, said that q restricted to the kernel of z ternzink R or zr is obtained by extension of scale, so it will be a map from gamma R which is Rn to C which is R2, so you get a map from Rn to R2 and on the kernel the form should be negative and q on gamma for any in if a gamma non equal to 0 then q of gamma is positive, so that's only one axon and now I think I can remove everything here, there is no rule. Sorry? Yeah? No, no, it's actually interesting, it's maybe related to this thing but it's a good question here. No, I don't and let me just define this abstract notion because the algebra is of very, very different nature. Yeah, for example if n equals 3 it's kind of easy to measure this function, so I just can see the cube, it will be just some part in R3 and I project to a plane, yeah, a square city kernel 2, it will be my subtle charge and here I get all integer points or some kind of lattice is my cube and what the support property means is the following, so there exists certain, here it will be two cones which can, with the axis, the square of my map, vertical line and so all this gammas which appears with the game are outside of these two cones and when you get a projection you get a discrete subset of R2 automatically because on each domain you get only finitely many things in between, so you get a subset with some kind of polynomial growth growing number in a disk when you exchange center radius and the order of growth depends on the rank of your lattice. So you get the things and if you go to proctivization I would like to think it's like this, you imagine this maybe subset, I don't know, like sphere S2, you consider some domain like this and take cone of it, so it will be very, very nonconvex cone where is your elements are present and what is wall crossing formalism, now start to rotate a little bit central charge and the claim is you can define canonically new wall cross, you can, you are assigned new elements more or less in the same domain and how to do it and in fact one can write in visual form but one can write pretty concrete thing like this, so suppose you have central charge let's do it in three dimensional and we project to a plane and divide my plane in several sectors and when we take pullback on the sectors it means that we divide my things on the sphere we divide by certain kind of like fogons and what will happen we get sectors V1, V2 and so on and for each sector we get corresponding part of my stability structure which lessens the sector and each VI gives cyclically ordered element AV which will be the following, it will be product increase argument of Z gamma, of gamma such that Z of gamma belongs to my sector and I take exponent of let's say A gamma, it will be element of certain pronular important group, important important group as such to some convex cone it should take convex hull of this guy if it's less than 180 degrees you can include some strict cone and get some pronular important group, so it's element and some pronular important group and if you subdivide the things the element will decompose in the product. Now how to recalculate the things if you rotate a little bit central charge, now we make a different projection, now kind of projection some kind of different way, the sphere to R2 and here we draw kind of completely different class of lines, when we take pullback we get something which is not, when we take pullbacks of this rays we get kind of hyper plane sections which are not parallel to, don't contain original cone of central charge but they stay in between and what we do, this element, this element correspond to each original sector, I'll just draw in a kind of a local picture is the following, I have this strip, I just go to universal cover I get decomposition like this and then some decomposition like this on different pieces by one projection to another. Are you assuming this hyper plane section is not parallel to the kernel of zeta or is by computation? Sorry? Are you assuming this hyper plane section is not parallel to the kernel of zeta, that's what you said? It's for the different zeta. It's new zeta, you get zeta prime close to zeta, close but not equal and now I can do the following, this element AVI, I can canonically uniquely decompose the product of elements where I use only this part of the algebra or this part of the algebra, kind of, so my element is decomposed by two parts and now what we can do, we'll multiply this by this and associate to new sector for new central charge. It will be element and then uniquely decompose the product of something in a clockwise order. So it's kind of define A prime whatever for this green guy as a product etc. How the uniqueness come, every time you say unique? You can uniquely decompose the element because the algebra splits in direct sum of two subalgebra and any element of the same important group can uniquely split in the first group and the element of the second group. So what was the green and blue lines? Green lines will be pull back of certain finite collection of rays for projection Z. Green lines projection of some kind of different rays maybe, basic traces of these angles, yeah, projection Z prime, yeah. Anything, it doesn't matter, the final answer does depend on what you do. So that's roughly of all crossing formalism, this can be done with kind of like effective computer program if you know in graded algebra how decompose element in subalgebra then it defines this new elements and in fact you don't even go to this complicated guys A gamma at all. You just have collection of elements AVI whose logarithms are supported in appropriate cones and then construct new group elements but you should alternatively use one or another decomposition on angle sectors. Yeah, okay, yeah, so it's a general idea what is going on. There's a question, Helen. The general conclusion is that the product doesn't change when you move the rays or what's the final conclusion? Yeah, it's kind of locally, what globally preserve it's hard to say. For example, if you get certain angle sectors for which there's no just on projection in complex in C it projects, there's some angle sectors where nothing goes on and there are two such sectors, the total composition will preserve and so what's people called spectrum generator and for DT in variance field this is canonical element. In the case of quivers it happens, you can say just one big element on some prongly important group stays the same but you decompose in different order. But what I'll talk about the situation when this projection is this kind of the directions of arguments of all this projections of this things if you consider rays they everywhere dance so this one big object is preserved. Okay. Does AVI have monodromic interpretation? What? Is there a monodromic interpretation of this product? AVI? Not really, no. And you see that also it's in kind of real life situations that's when you get central charge is given by rational numbers, this boundaries of the curve some irrational numbers, there's really no good choice and that's a little bit of trouble. You can expect to be some algebraic transformations. What else is it in kind of good situations in this hyper plane irrational this transformation could have some special property to be algebraic or rational and so on but otherwise it's some complicated guys. Okay. So now what will be the algebras I told you that for this counting of saddle connections there are two formalisms and there are the algebras so the lattice is for quadratic differentials. Differential is simple zeros so this lattice gamma is if I analyze it's maybe first homology of my spectral curve and we take 19 variant part under evolution for this. Yeah, this can be abstract definition of this lattice. Now get lattice with a scalar product. This projection, so what is the evolution idea? On a spectral curve it's double cover of your original curve so this is an evolution here. And but for any so it gets map from lattice to z and then you get a Lie algebra G gamma and then G gamma will be one dimensional for A E G gamma and this algebra Lie algebra will be algebra of function on algebraic torus with some Poisson bracket so roughly it has some I will do it has some base element and E gamma 1 bracket with gamma 2 will be scalar product of gamma 1 gamma 2 multiplied by E gamma 1 plus gamma 2. In fact there are some minus sign like here but I don't want one can rescale them and we can disappear. So it's algebra of Poisson vector field algebraic Poisson vector fields on the torus. And what I'll tell you right now it will be different story. Suppose have another curve C and now a billion differential not quadratic differential and a node by alpha. Maybe a curve even curve should not by different curve it's kind of different curve and strictly speaking it's my spectral curve maybe it's not my sigma curve. If you compare to this situation on the double cover get one form. Suppose have a billion differential and my group will be h1 of curve 0 of alpha with a slightly different group and the central charge is given by integral of alpha. So you are considering a billion differential of first type or any type second type along in singularities. One can consider actually arbitrary, meromorphic, a billion and the algebra will be different. And speaking it will be matrix valued functions on algebraic curve on the torus which is again different and matrices of certain size roughly size number of zeros. But what I claim so you get matrix valued function and this identity will hold like for matrix valued functions on the torus. At each point you will get similar identity when you change get situation like this. And in fact the whole story works in more general. You consider any let's say compact complex even it can be addressed infinitely far but let's assume compact complex manifold of any dimension let's say dimension manifold will be say m. Emical one case of curve. And suppose you get helomorphic one form which is closed. And the whole story very general and suppose it has isolated zeros. And now I'll make something which can break me continue to assume that zeros are simple zeros. It goes to the general to non-simple zeros which doesn't correspond to this story to double cover because when you go to double cover here you get quadratic differential has something like singularities z, dz squared. If you go to double cover, right z is w squared, z is squared of w, yeah. Go to double cover then I get something like w squared dw. On double cover get double zero not simple zero. It's additional complication here it's very easy to work out what we're going for double zero but I will explain the simple zero so it's not directly applicable to the previous story. Yeah, so you get this very even high dimensional variety. I claim that one get immediately some more crossing formulas. The claim is the following. Where does all crossing formulas come from? The idea is the following if you have this new variety and one form alpha here now one can let's take real part of one form get real one form close real one form. This will be close to real one form. And now one can try to do the following. This is close real one form. You want to present as a differential of some function. It's impossible because form has some periods. So you go to universal abelian covering of your x and then back will be differential of some function. And this function will be function with more singularities of index m. m is dimension of my manifold because when you get complex more I assume the form has a simple zero so it means it's locally I can represent complex one form is differential of holomorphic function. It should be sum of squares but then of complex variables when I take a real part I get sum of m squares minus sum of another squares. So if you get something like z1 square plus zm square take real part and zi is xi plus yi is the same sum of xi squares minus sum of yi squares. So you get function on huge kind of very non-compact manifold with all more singularities of the same index. Now pick any Hermitian metric not even kerimetic on x and it gives a gradient flow. So what goes on your manifold you get on universal cover you get kind of like infinitely many physical points because consider zero of my form and go to universal cover get many copies and then for each one form one get certain by gradient line we get certain let's say unstable manifold will be copy of RM will be unstable manifold as such to this more syntax because they have the same more syntax you expect the things do not intersect each other and the fact is the following assume that the integral of my holomorphic one form for any path in connected to singular points and get set of zeros I consider any path with maybe the center mark of points it doesn't belongs to positive number. That's typical this generic condition because this is lattice of finite rank and if it's a wrong you just multiply alpha by generic not rotate by generic angle and then this countable subset will not go to positive numbers and then this six is this left shift symbols are disjoint there will be no gradient line kinetic one guy to another at all because along this gradient line it's actually derivative real part if r is real part of some complex function and by this condition because it's Hermitian metric this imaginary part of fc stays the same stays constant along the gradient line yeah if you say that it's zero then we see that the integral fun form will be strictly positive number because imaginary part will be zero and real part will be integral of positive guy it will look strictly positive so it contradicts this condition so you see there's no left shift symbols and you get this story yeah so you get you get it looks like you get a lot of homology classes but these homology classes are not really compact one and what one can do it's some kind of part of this Novikov theory of this more snorkel complex that's left shift symbols give homology classes in what in the homology of your original manifold with coefficients in some local systems L is a local system of rank one and what I want for this local system this if you go further along this left shift symbols then is the size of your vector should decrease so strictly speaking one should consider L over some non-archimedean field and when you get with the properties that when you transfer this following property if you have a local system is classified by h1 with coefficients in the vertebral element of your local field to consider logarithm of norm get class in h1r and then should be represented by some one form such that b pairing with gradient flow is positive outside of zeros of your gradient flow and then if you have this local system and if you make parallel transport along this gradient flow you get small and small things and all together you get kind of like chain m dimensional chain so this element will be classes in hm and there will be no other co-homology so in this case it says that for such local system all commodules will be concentrated in one degree and the space will have certain bases now one can do kind of more final calculation I want to show you some situation how one now what should I do now I may be introduced some parameter theta I start to use different flows and so you get a basis in homology or homology say x is local system l local system so class l in h1 of my field xk star and it belongs to certain domain open domain in any con direction and not for all local system but some domain I get a basis depending on on theta because I can start to can rotate and ceta should be generic ceta is kind of irrational ceta is not equal to argument of central charge of gamma for any gamma in my lattice which is h1 of pair x0 alpha I get a basis now I take two elements and I compare two basis elements to two bases I get transformation from one basis to another and what I get I get a matrix valued function on on the stores in size of matrices will be number of zeros and obviously if we divide I get composition and if I this two things very close to each other then it's easy to see that I get elements corresponding to the algebra corresponding slope so I get wall crossing structure and the next things if I change complex structure one form I get this all this for my formalism gives you recalculated things want to know so if you know it at one point I can with this formal manipulation calculate for different complex manifold when I deform complex structure in one form and so the one I have done only one calculation in this kind of very simple example how to create this change of basis doing from one direction to another and now I'll draw a bit long yes if you make notes just try to draw things efficiently large because these things will kind of grow a picture from itself now so my so special case x will be surface of genus zero genus zero curve and I choose one form which has two zeros two simple zeros and periods belong to Z plus square root of minus one Z and in fact it will be the circle will be composed in four squares it's kind of minimal possible example in this case things I have four squares now I'll just draw you this four squares so surface is good from this so square some squares it's in C so this like standard coordinates DZ will be this my form will be standard chords Z now should glue the edges yeah in fact I'll have something like so the composition yeah so the glue edges so this will be E1 glue with this E1 it will be first stage from here to here I should identify somehow edges the composing the pair how I glue an arbitrary way so it will be here here and now glue it back to three here if for itself and now for vertical edges will be something similar six here will be six here seven here will be the same as seven here so from opposite side and here to be maybe eight yeah so we have four squares will be eight edges and then this guy will have also two vertices if you start to glue boundary vertices you get kind of like V1 V1 and get second vertex V2 which is okay so you get plenty of vertices now should write local system let's say trivialized at my zeros I consider local systems trivialized trivialized at this is my two vertices so it means that for each edge I should write what is the whole of my system so that's it has five parameters it will be a first commode of pair will be five parameters gamma it will be five and then one can write which call it to choose case like this one can prioritize this monoramy in this way so a b f gt will be five parameters here so my torso the Laurent polynomials in these five variables okay so as you get this sink now I should draw so that's now I can write chain complex for calculating each one of my surface access coefficient the local system depending on this parameters you know by FGT yeah I just write you just to see the idea what is going on sorry it was some sick music in my picture yeah for example one when one calculate boundary of let's see e1 this guy it will be two minus from this end minus e v2 from another end and so on so you can write what is the boundary of differential or for example you consider square one this first cell the boundary will be the following guide will be e5 plus b gt times e1 because to reach e1 I should go through this things minus e3 plus ie a times e5 it's e3 and to reach a5 I should go through this monoramy get collection by a yeah so you get this you can write immediately this chain complex and then first homology will have rank two now I want right in this complex certain elements when I put here this infinite chain this left-shift symbol and what is left-shift symbol I recall that we get this one form can make what form it gives me some kind of this rented relation and what I should do I should make this will be my left-shift symbols because it will project in under integration to it's double cover of array symbol it's in a first thing symbol yeah okay so this is its left-shift symbol it's a left-shift symbol associated some point I it will be difference of two trajectories as a rented cycles it's one thing is this direction plus another oriented leaf of relation is opposite orientation it will be kind of symbol plus vi minus symbol minus vi now I want to yeah but to draw the whole things I should go to vertically horizontal direction and maybe I just say you I use as a direction theta it will be zero degrees plus very very small kind of like zero point infinitesimally small positive direction because it's rational angle I should leave a little bit to rational so it will be kind of limiting picture so what does it mean I should start to draw from your text V1 I start to draw things slightly going like this it goes to V5 and then it's continuous it's returned to itself and then continues itself itself itself so it will be one of my this because this side is glued to this side if I go from here it's returned I start to rotate infinitely many times so I get some some kind of like this things yeah it's will be just one of half and then another thing that should start from vertex V1 but in different square also continue here I entered E8 so then I continue here from a six I continue here from seven continue here and so on so it will be kind of another thing and one can write the formulas for example for this gamma of vertex V1 plus this will be plus and this will be one will be very simple it will be what it will be h e3 plus a times e3 let's say square times e3 and so on here because you go around around around what is it I get one over minus a times e3 and gamma V1 minus it will be more complicated some yeah for example I consider this e2 I start from here then I reach my six a8 then it should at least at times next guy e1 plus a square t times e4 plus a cube f times e2 and so on yeah so things kind of start to repeat after factor to the sort of term so and we make some arithmetic progression get one minus one a cube f e2 plus a t times one minus f e1 plus a square t times one minus a cube f e4 yeah so it will be the final sum so you get some rational expression this plus this so that one can make the second base element get base element express sources only horizontal lines now I can do vertical things get another basis and come all to a different base now this change of coordinates change of coordinates it's still more complicated metrics so as a final formula is like this sorry a bit long but this is really one final formula and eventually one can make complete computer and get how do you note my basis gamma sorry gamma Vi okay yeah yeah I can see the vertical directions will be kind of 90 degrees it will be not exactly 90 degrees but 80 89 999 degrees and horizontal be zero just like this so it will be certain metrics applied in and the matrix is the following guy this is the last formula which took me some maybe one minute to write here we get keys times some another one straws expression yeah Just second let's finish inner glass Okay, that's it. Okay, so I get this pretty explicit rational value of matrix value of function. The rationality is clear because I use some of geometric progressions, the rational things. And now one should use some kind of central charge for this monomials. The central charge of monomials will be the following. Central charge of monomials A will be plus one. Central charge of monomials B will be plus E. And central charge of all the rest will be zero. Yeah, it's very degenerate, central charge. And it's responsible for us only for the kind of, it's not described as a full cross-structure because what I describe, I describe you go from almost vertical to almost horizontal piece. Yeah, it will be contribution for this limiting sector. In fact, one should use eight different sectors. There'll be infinitesimal goes to vertical and left and right. One should write actually eight different matrices to get the whole description. But then the claim, if you start, rotate, change a little bit your central charge. And then try to decompose in linearity piece. You see exactly number of this gradient lines on generic run to a generous to surface. Coming from this monstrous algebraic function. Yeah, so it's a different story. Yeah, which one can check numerically. It's definitely gives the right things. But I didn't relate yet to this work by Bridgman Smith because I explained that here I have simple zeros for my billion differential. But if you come from quadratic, you get double zeros. And what happens in this double zeros, it's the following thing in this gradient lines. You have a pitch critical point will produce not one cycle, but two linearly independent cycles. So the matrices you'll have again certain size. But just not basis, but identification with certain canonical small dimension of like vanishing commulsion vanishing cycles. But the whole story is super duper general and works for complex manifold of any dimensions, if not translated zeros, not compact one. Yeah, so it's really different story. But for most the same numbers. Yeah, okay, thank you. Sorry, no, this physics terminology, it's some, there's some lattice and this is some extension of some super algebra and there'll be some super symmetry. No, central church, it's a functional on a lattice or element of some finite dimensional vector space. And your algebra is just Hamiltonian vector fields and torus. Yeah, no, there are two different store either Hamiltonian vector fields and torus or matrix valued functions on the torus. Yeah, so it will be like gravity and gauge theory, I would say. Yeah, because it's in the, in the de-phorophysm and. What is the type of matrix and this? It's two by two matrix. No, any specific like any. Sorry? Does it belong to some kind of real algebra or something? No, of course it's a, it's a, matrix belongs to two by two matrices. This coefficients in the what? Yeah, in fact, it will be series in AB. In AB, these coefficients in maybe some polynomials in F, J and T. Somewhere, okay, some such this. It's all the form of power series. You expand A, B. Is it this consequence of the central charges of F, G and T is zero? Yeah, it's kind of a very special one form. Yeah, you make it comes from some homology classes. FG is built to some homology class and to get one form, we get this, this numbers. There is an online question. Could one insert a spectral parameter in this formula with the T, with the T's to deform the relation to make it look like a young backstreet. I don't know. It's a three. Yeah, it's a three. Yeah, it's a three. It's different. It's not two equal to three. So I have a question. T in this formula seems a little bit different from the others. Yes, yes, T, it's just it's kind of AB, AB, FG are really a model of local systems and T responsible for comparison of trivialization of two points. So it's a matrix that just appears only like a very special way. So taking T equals zero, it's normal. No, T equal one. Yeah, so T equal one is the right limit here. Any other questions? It doesn't appear in the determinant actually. Yes, yes, the term to the matrix is one. You can calculate it. It's very, just multiply it. Yeah, in fact, without computer, I wouldn't be able to write this formula. I have to say it's. So here you fixed the Chinese two curve, you know, to do this competition. Do you have a monotony or you change the curve? Yeah, no, no, you start to change the curve. You change the central charge. And then you really decompose this stuff. Yeah, changing the stability. Yeah, change stability. Yeah, here I have no idea what is kind of category theoretic interpretation of this. Or maybe I have, but I, but I. Yeah, here it's three spheres, but this is the whole story. It's in any dimension. It's not three dimensional yet. Just just for illustration. Are there any questions? It's not the three spheres in this top bundle you are making. No, no, no, here I think I do do something different. I go to cotangent bundle to my curve. I have zero section and graph graph of my one form. And then for category, I can see the X identify with local and then it's something in this direction. So I go to Foucaille category of cotangent bundle to this curve, which is spectral curve in Bridgian Smith. Sorry. It's time to watch again. Thank you.
 Thank you very much for the invitation.  So the topic of my talk will be slightly different from what you heard before.  It will be kind of new direction in wall crossing.  But some topic, in fact, it wasn't really addressed on this conference about wall crossing  for three-dimensional Calabio categories in general.  Yeah, so this story which we started with Jan quite a long time ago.  So the idea is that you have triangulated category and some technical properties we  call inconstructible objects form.  Can be characterized by algebraic varieties, which is three-dimensional Calabio, and when  we have stability structure, then it should produce certain DT invariance, certain integers,  variable by some indices, and changing stability structure DT invariance by some certain wall  crossing formulas.  Yeah, so it's a whole story.  But people in algebraic geometry, example of this, like the following three Calabio  categories.  Examples of categories from algebraic geometry are the following.  You take three-dimensional complex manifold, algebraic variety, say, this canonical class  trivial.  This could be compact or not compact.  And now you choose, pick some certain subset Y in X cube, it will be compact algebraic  subset, and then one consider a category of perfect complexes on X with supported Y.  Y could be a point curve, or could be Y itself, it's Y is compact.  And in a reasonable situation, one can expect there will be some stability structures like  Gizikir stability, and then we count roughly complex coherence sheets of stable pairs with  support on Y, and get certain guys, and then we can recalculate.  And I think Richard Thomas gave you some examples for such situation.  We can get some Y as a surface, and X as a total space of canonical bundle on the surface.  In algebraic geometry, it's pretty hard to construct the stability structures, and there  is kind of mirror-dual picture.  You consider some kind of six-dimensional, real symplectic, infinity-symplectic manifold,  and flex-dual, because it's a real source of six-dimensional, maybe not compact one.  And in fact, you also assume that it has complex structure, and canonical class trivial, and  it has holomorphic reform, and objects which you consider will be, now consider something  called Foucaille category of your variety.  So roughly consider Lagrangian submanifers, but nobody knows exactly what is it.  And then stable objects in this picture should be special Lagrangian submanifers.  So it means it's Lagrangian submanifers, such as restriction of this 3-0 form has this constant  argument, which I denote by theta.  Yeah, here's a kind of different situation.  We don't really know what is the category, but kind of know, up-rower, what are stable  objects.  Here we know the category, but stable object, it's hard to see.  And these categories are not really constructed rigorously in any concrete case, but there  are some things which we can guess what is going on.  Namely, there was a story constructed by Tom Bridgeland and Ivan Smith.  They constructed some category with stability structure looking on this picture.  They don't directly identify it with Foucaille category, but obviously it comes from this  picture.  What they started with?  They started with some non-compact algebraic curve and with quadratic differential.  So this small cube will be section of curve and square of canonical bundle.  The quadratic differential with simple zeros.  And they have some numerical conditions, so the curve should have sufficiently many  zeros and the form of quadratic differential should have full infinity.  It's expected this should work for compact curve, but I don't think it's really constructed  this category in this case.  So what you do with this curve?  Morally, you construct a, you do the following.  If you consider cotangent bundle to the curve, which is a very complex dimensional variety,  and it contained spectral curve given by quadratic differential.  It will be plus minus square root of quadratic differential gives you a billion differential  defined up to two valued section of these things.  And it will be smooth curve, spectral curve to be kind of doubly ramified at zeros of  your quadratic differential.  And yeah, it's complex two dimensional variety and now I construct something like called  conic bundle.  So the bundle which have fiber C star outside spectral curve and C union C can be degenerate  hyperbola over spectral curve.  This will be three dimensional variety.  And then this three dimensional variety one try to see what are special Lagrangians.  Yeah, so you don't really think about what is scalar form, but one can guess what are  special Lagrangians and the idea is that they can respond to one to one to the following  things.  It's called, what's called saddle connections.  Make very, and close draw basics.  What do I mean?  If you have quadratic differential, then it get, take absolute value to get a flat metric.  But with conic singularities because it's zero quadratic differential metric is not defined.  It's conic singularities and conic singularities, the total angle will be three pi instead of  two pi.  Oh, sorry, no, sorry, conic singularities you get three pi, yeah.  That's the angle.  And so it gets a flat metric and then you can be interested in just kind of geodesic  lines containing, connecting to zeros.  How one can proceed, how this one can associate, this saddle connection can associate a sphere  sitting in this three dimensional Calabio.  How we do?  If you got saddle connection, yeah, so this is your curve and here's a double cover spectral  curve.  First of all, what you'll construct, you construct a disk in a total space, namely for each point  on the saddle connection, you consider two points of the spectral curve and connect by  real interval.  Because we get double cover of this thing, you get small interval connection, just two  choices of square root, which degenerate to a point on the boundary, so you get a disk,  you get two dimensional disk.  This boundary, yeah, so from these things you get two dimensional disk in the cotangent  bundle, as I did boundary, leaves in the spectral curve.  Okay, now I have a bundle with fiber C star and the C star get middle circle.  So I can see the bundle with fiber middle circle of interior of the disk, degenerate  to a point and I get three dimensional sphere, oh, sorry, what, yeah, yeah, this way I get  a three dimensional sphere.  Okay, and one can convince yourself if you put appropriate very degenerate symplectic  structures, it will be really special agrarian.  And similar if you get closed geodesics, closed geodesics you get your S1 crosses 2.  See you can see three dimensional manifold.  Yeah, okay, so you got this hypothetical picture and what they really constructed, they constructed  categories and these are objects and this will be simple stable objects.  And construct stability condition, yeah, that's complicated work.  And then by general think which is also very complicated to get full crossing formulas,  but the composition of the thing is really elementary.  So what here goes on, I get certain lattice which is H3 of this three kalabeya which constructed  with integer coefficients.  It has intersection form, skew symmetric form.  And then I get a map from the slattest to complex numbers.  Yeah, there are many ways to see it, for example, the slattest can be identified with H2 of  cotangent bundle to my curve relative to spectral curve is integer coefficients.  And then you integrate canonical one form PDQ on cotangent bundle.  So get this functional and you get, and this whole crossing formulas, very abstract whole  crossing formulas is coming from any three kalabeya categories to any lattice and the  skew symmetric form.  And DT invariance, they have contribution of two types, namely saddle connection, which  gives me a sphere and then class or it gives you a certain class in this lattice and it  gives, and this number will be one because it will be isolated sphere in just one.  And for closed geodesics, for closed geodesics, then it's slightly different because closed  geodesics, they automatically appear in one parameter family, which relate to the fact  that it's one crosses two, it has first co-emulsion, it can move.  So get one parameter family and, strictly speaking, what will cause on the following.  You should consider not only Lagrange manifold, but rank one local systems, maybe this gives  you non-recommendant field.  So you get something like annulus in some non-recommendant field.  Yeah, so what you form annulus of some non-recommendant field, but this cylinder has some kind of  end points and at end points something complicated goes on the 4k category and this thing's kind  of like compactified to project space to P1 and then it gives omega of annulus of closed  geodesics will be plus two, the earlier characteristic of P1.  Yeah, so it's some kind of explanation, but one can sit at the rest of the receipt, take  number plus one and plus two for this guys and then if you start to change your quadratic  differential on a curve, it means that you change a little bit stability structure on  your things, these things should be recalculated in certain way.  And here's some kind of basic identity, not dramatic, and this, so you get certain identity  how the numbers are calculated and for this you don't need all the whole machine, you  can do it elementary by hand, just drawing geodesics and gradient lines on a curve.  And for example what happens typically, you have maybe three zeros and you get one geodesic  and another geodesics, but here you get two angles alpha and beta and alpha plus beta is  equal to three pi, but when you change complex structure in the form and then if alpha is  equal to pi and beta is still very large to pi, then in this moment you'll have another  geodesic connecting, if its angle is really less than pi, so you get a triangle and then  you have another geodesic which appear.  So you go from two geodesic, two saddle connections, goes to three saddle connections.  So this is typical transformation what is going on and here's this wall crossing identity  kind of famous in this thing which is, you've seen it many disguise, so you get identity,  corresponding identity with certain elements in certain group which will be simplectomorphism  in two variables, namely you can see that for any integers a and b, you can see the  transformation just in two variables, variational transformation which will be clustered transformation  x multiplied by one minus, you get transformation from variational transformation for plane to  itself and then you get this identity, of course it's 0, 1 you get essentially just  two choices for one, one will be one sign, one 0 will be different sign and all is based  essentially on this identity and you also analyze something which happens with these  cylinders and will be another infinite identity for such guys and how we can prove this wall  crossing formula directly as going to this row to categories and this complicated machinery  with whole story, so it's completely elementary.  In fact it was known from many years there was something similar goes on for much simpler  group, here the group is group called variational transformation of plane but now consider just  three by three matrices, you can put three by three matrices and then there is similar  identity namely one, one, one, one, one, all the rest is 0, you get this identity for three  by three matrices and if you take a logarithm, get elements in some graded algebra, yeah  it's for many years I was puzzled why there are such two completely different groups,  they have inside this kind of bunch of elements, it's the same identity and the question, can  one see this counting of subtle connections through this perspective, not through this  wall crossing perspective and what the goal of my talk to explain is yes, there is different  wall crossing formulas for counting almost the same subtle connection, I do not count  closed geodesics now anymore, I consider count only subtle connections and there will be  this wall crossing formalism which works for them, yeah so it's pretty surprising, you  get two different groups and the products and the compositions are essentially the same,  so and the second story, maybe I'll just tell you a few words what are all this wall crossing  formulaic, what does a wall crossing formalism is in general, it starts with, it's almost  nothing, so we have let's say a lattice of finite rank, like z to power n and suppose  we have a Lie algebra, let's say overrational numbers which is gamma graded and so the commutator  of the homogeneous element is homogeneous of degrees equal to the sum, just say Lie  algebra and what I call stability structure on G is the following, is there is z and a  collection of a gamma, z is a map from my lattice to complex numbers with additive map,  so just enough to define on the basis of the lattice and a gamma is a collection of elements  but for nonzero elements of the lattice, so it is saying only one stupid axon is called  support property which is the following, there exists the quadratic form on gamma cross  R, said that q restricted to the kernel of z ternzink R or zr is obtained by extension  of scale, so it will be a map from gamma R which is Rn to C which is R2, so you get  a map from Rn to R2 and on the kernel the form should be negative and q on gamma for  any in if a gamma non equal to 0 then q of gamma is positive, so that's only one axon  and now I think I can remove everything here, there is no rule.  Sorry?  Yeah?  No, no, it's actually interesting, it's maybe related to this thing but it's a good  question here.  No, I don't and let me just define this abstract notion because the algebra is of very, very  different nature.  Yeah, for example if n equals 3 it's kind of easy to measure this function, so I just  can see the cube, it will be just some part in R3 and I project to a plane, yeah, a square  city kernel 2, it will be my subtle charge and here I get all integer points or some  kind of lattice is my cube and what the support property means is the following, so there  exists certain, here it will be two cones which can, with the axis, the square of my  map, vertical line and so all this gammas which appears with the game are outside of  these two cones and when you get a projection you get a discrete subset of R2 automatically  because on each domain you get only finitely many things in between, so you get a subset  with some kind of polynomial growth growing number in a disk when you exchange center  radius and the order of growth depends on the rank of your lattice.  So you get the things and if you go to proctivization I would like to think it's like this, you  imagine this maybe subset, I don't know, like sphere S2, you consider some domain like this  and take cone of it, so it will be very, very nonconvex cone where is your elements are present  and what is wall crossing formalism, now start to rotate a little bit central charge and  the claim is you can define canonically new wall cross, you can, you are assigned new  elements more or less in the same domain and how to do it and in fact one can write in  visual form but one can write pretty concrete thing like this, so suppose you have central  charge let's do it in three dimensional and we project to a plane and divide my plane  in several sectors and when we take pullback on the sectors it means that we divide my  things on the sphere we divide by certain kind of like fogons and what will happen  we get sectors V1, V2 and so on and for each sector we get corresponding part of my stability  structure which lessens the sector and each VI gives cyclically ordered element AV which  will be the following, it will be product increase argument of Z gamma, of gamma such  that Z of gamma belongs to my sector and I take exponent of let's say A gamma, it will  be element of certain pronular important group, important important group as such to some  convex cone it should take convex hull of this guy if it's less than 180 degrees you  can include some strict cone and get some pronular important group, so it's element  and some pronular important group and if you subdivide the things the element will decompose  in the product.  Now how to recalculate the things if you rotate a little bit central charge, now we make a  different projection, now kind of projection some kind of different way, the sphere to  R2 and here we draw kind of completely different class of lines, when we take pullback we get  something which is not, when we take pullbacks of this rays we get kind of hyper plane sections  which are not parallel to, don't contain original cone of central charge but they stay in between  and what we do, this element, this element correspond to each original sector, I'll just  draw in a kind of a local picture is the following, I have this strip, I just go to universal  cover I get decomposition like this and then some decomposition like this on different  pieces by one projection to another.  Are you assuming this hyper plane section is not parallel to the kernel of zeta or is  by computation?  Sorry?  Are you assuming this hyper plane section is not parallel to the kernel of zeta, that's  what you said?  It's for the different zeta.  It's new zeta, you get zeta prime close to zeta, close but not equal and now I can do  the following, this element AVI, I can canonically uniquely decompose the product of elements  where I use only this part of the algebra or this part of the algebra, kind of, so my  element is decomposed by two parts and now what we can do, we'll multiply this by this  and associate to new sector for new central charge.  It will be element and then uniquely decompose the product of something in a clockwise order.  So it's kind of define A prime whatever for this green guy as a product etc.  How the uniqueness come, every time you say unique?  You can uniquely decompose the element because the algebra splits in direct sum of two subalgebra  and any element of the same important group can uniquely split in the first group and  the element of the second group.  So what was the green and blue lines?  Green lines will be pull back of certain finite collection of rays for projection Z.  Green lines projection of some kind of different rays maybe, basic traces of these angles,  yeah, projection Z prime, yeah.  Anything, it doesn't matter, the final answer does depend on what you do.  So that's roughly of all crossing formalism, this can be done with kind of like effective  computer program if you know in graded algebra how decompose element in subalgebra then it  defines this new elements and in fact you don't even go to this complicated guys A gamma  at all.  You just have collection of elements AVI whose logarithms are supported in appropriate  cones and then construct new group elements but you should alternatively use one or another  decomposition on angle sectors.  Yeah, okay, yeah, so it's a general idea what is going on.  There's a question, Helen.  The general conclusion is that the product doesn't change when you move the rays or what's  the final conclusion?  Yeah, it's kind of locally, what globally preserve it's hard to say.  For example, if you get certain angle sectors for which there's no just on projection in  complex in C it projects, there's some angle sectors where nothing goes on and there are  two such sectors, the total composition will preserve and so what's people called spectrum  generator and for DT in variance field this is canonical element.  In the case of quivers it happens, you can say just one big element on some prongly important  group stays the same but you decompose in different order.  But what I'll talk about the situation when this projection is this kind of the directions  of arguments of all this projections of this things if you consider rays they everywhere  dance so this one big object is preserved.  Okay.  Does AVI have monodromic interpretation?  What?  Is there a monodromic interpretation of this product?  AVI?  Not really, no.  And you see that also it's in kind of real life situations that's when you get central  charge is given by rational numbers, this boundaries of the curve some irrational numbers, there's  really no good choice and that's a little bit of trouble.  You can expect to be some algebraic transformations.  What else is it in kind of good situations in this hyper plane irrational this transformation  could have some special property to be algebraic or rational and so on but otherwise it's some  complicated guys.  Okay.  So now what will be the algebras I told you that for this counting of saddle connections  there are two formalisms and there are the algebras so the lattice is for quadratic differentials.  Differential is simple zeros so this lattice gamma is if I analyze it's maybe first homology  of my spectral curve and we take 19 variant part under evolution for this.  Yeah, this can be abstract definition of this lattice.  Now get lattice with a scalar product.  This projection, so what is the evolution idea?  On a spectral curve it's double cover of your original curve so this is an evolution here.  And but for any so it gets map from lattice to z and then you get a Lie algebra G gamma  and then G gamma will be one dimensional for A E G gamma and this algebra Lie algebra  will be algebra of function on algebraic torus with some Poisson bracket so roughly  it has some I will do it has some base element and E gamma 1 bracket with gamma 2 will be  scalar product of gamma 1 gamma 2 multiplied by E gamma 1 plus gamma 2.  In fact there are some minus sign like here but I don't want one can rescale them and  we can disappear.  So it's algebra of Poisson vector field algebraic Poisson vector fields on the torus.  And what I'll tell you right now it will be different story.  Suppose have another curve C and now a billion differential not quadratic differential and  a node by alpha.  Maybe a curve even curve should not by different curve it's kind of different curve and strictly  speaking it's my spectral curve maybe it's not my sigma curve.  If you compare to this situation on the double cover get one form.  Suppose have a billion differential and my group will be h1 of curve 0 of alpha with  a slightly different group and the central charge is given by integral of alpha.  So you are considering a billion differential of first type or any type second type along  in singularities.  One can consider actually arbitrary, meromorphic, a billion and the algebra will be different.  And speaking it will be matrix valued functions on algebraic curve on the torus which is again  different and matrices of certain size roughly size number of zeros.  But what I claim so you get matrix valued function and this identity will hold like  for matrix valued functions on the torus.  At each point you will get similar identity when you change get situation like this.  And in fact the whole story works in more general.  You consider any let's say compact complex even it can be addressed infinitely far but  let's assume compact complex manifold of any dimension let's say dimension manifold will  be say m.  Emical one case of curve.  And suppose you get helomorphic one form which is closed.  And the whole story very general and suppose it has isolated zeros.  And now I'll make something which can break me continue to assume that zeros are simple  zeros.  It goes to the general to non-simple zeros which doesn't correspond to this story to  double cover because when you go to double cover here you get quadratic differential  has something like singularities z, dz squared.  If you go to double cover, right z is w squared, z is squared of w, yeah.  Go to double cover then I get something like w squared dw.  On double cover get double zero not simple zero.  It's additional complication here it's very easy to work out what we're going for double  zero but I will explain the simple zero so it's not directly applicable to the previous  story.  Yeah, so you get this very even high dimensional variety.  I claim that one get immediately some more crossing formulas.  The claim is the following.  Where does all crossing formulas come from?  The idea is the following if you have this new variety and one form alpha here now one  can let's take real part of one form get real one form close real one form.  This will be close to real one form.  And now one can try to do the following.  This is close real one form.  You want to present as a differential of some function.  It's impossible because form has some periods.  So you go to universal abelian covering of your x and then back will be differential  of some function.  And this function will be function with more singularities of index m.  m is dimension of my manifold because when you get complex more I assume the form has  a simple zero so it means it's locally I can represent complex one form is differential  of holomorphic function.  It should be sum of squares but then of complex variables when I take a real part I get sum  of m squares minus sum of another squares.  So if you get something like z1 square plus zm square take real part and zi is xi plus  yi is the same sum of xi squares minus sum of yi squares.  So you get function on huge kind of very non-compact manifold with all more singularities of the  same index.  Now pick any Hermitian metric not even kerimetic on x and it gives a gradient flow.  So what goes on your manifold you get on universal cover you get kind of like infinitely many  physical points because consider zero of my form and go to universal cover get many copies  and then for each one form one get certain by gradient line we get certain let's say  unstable manifold will be copy of RM will be unstable manifold as such to this more  syntax because they have the same more syntax you expect the things do not intersect each  other and the fact is the following assume that the integral of my holomorphic one form  for any path in connected to singular points and get set of zeros I consider any path with  maybe the center mark of points it doesn't belongs to positive number.  That's typical this generic condition because this is lattice of finite rank and if it's  a wrong you just multiply alpha by generic not rotate by generic angle and then this  countable subset will not go to positive numbers and then this six is this left shift symbols  are disjoint there will be no gradient line kinetic one guy to another at all because  along this gradient line it's actually derivative real part if r is real part of some complex  function and by this condition because it's Hermitian metric this imaginary part of fc  stays the same stays constant along the gradient line yeah if you say that it's zero then  we see that the integral fun form will be strictly positive number because imaginary  part will be zero and real part will be integral of positive guy it will look strictly positive  so it contradicts this condition so you see there's no left shift symbols and you get  this story yeah so you get you get it looks like you get a lot of homology classes but  these homology classes are not really compact one and what one can do it's some kind of  part of this Novikov theory of this more snorkel complex that's left shift symbols give homology  classes in what in the homology of your original manifold with coefficients in some local systems  L is a local system of rank one and what I want for this local system this if you go  further along this left shift symbols then is the size of your vector should decrease  so strictly speaking one should consider L over some non-archimedean field and when you  get with the properties that when you transfer this following property if you have a local  system is classified by h1 with coefficients in the vertebral element of your local field  to consider logarithm of norm get class in h1r and then should be represented by some  one form such that b pairing with gradient flow is positive outside of zeros of your  gradient flow and then if you have this local system and if you make parallel transport along  this gradient flow you get small and small things and all together you get kind of like  chain m dimensional chain so this element will be classes in hm and there will be no  other co-homology so in this case it says that for such local system all commodules  will be concentrated in one degree and the space will have certain bases now one can  do kind of more final calculation I want to show you some situation how one now what  should I do now  I may be introduced some parameter theta I start to use different flows and so you get  a basis in homology or homology say x is local system l local system so class l in h1 of  my field xk star and it belongs to certain domain open domain in any con direction and  not for all local system but some domain I get a basis depending on on theta because  I can start to can rotate and ceta should be generic ceta is kind of irrational ceta  is not equal to argument of central charge of gamma for any gamma in my lattice which  is h1 of pair x0 alpha I get a basis now I take two elements and I compare two basis  elements to two bases I get transformation from one basis to another and what I get I  get a matrix valued function on on the stores in size of matrices will be number of zeros  and obviously if we divide I get composition and if I this two things very close to each  other then it's easy to see that I get elements corresponding to the algebra corresponding  slope so I get wall crossing structure and the next things if I change complex structure  one form I get this all this for my formalism gives you recalculated things want to know  so if you know it at one point I can with this formal manipulation calculate for different  complex manifold when I deform complex structure in one form and so the one I have done only  one calculation in this kind of very simple example how to create this change of basis  doing from one direction to another and now I'll draw a bit long yes if you make notes  just try to draw things efficiently large because these things will kind of grow a picture  from itself now so my so special case x will be surface of genus zero genus zero curve  and I choose one form which has two zeros two simple zeros and periods belong to Z plus  square root of minus one Z and in fact it will be the circle will be composed in four  squares it's kind of minimal possible example in this case things I have four squares now  I'll just draw you this four squares so surface is good from this so square some squares it's  in C so this like standard coordinates DZ will be this my form will be standard chords  Z now should glue the edges yeah in fact I'll have something like so the composition yeah  so the glue edges so this will be E1 glue with this E1 it will be first stage from here  to here I should identify somehow edges the composing the pair how I glue an arbitrary  way so it will be here here and now glue it back to three here if for itself and now for  vertical edges will be something similar six here will be six here seven here will be the  same as seven here so from opposite side and here to be maybe eight yeah so we have four  squares will be eight edges and then this guy will have also two vertices if you start  to glue boundary vertices you get kind of like V1 V1 and get second vertex V2 which is  okay so you get plenty of vertices now should write local system let's say trivialized at  my zeros I consider local systems trivialized trivialized at this is my two vertices so  it means that for each edge I should write what is the whole of my system so that's it  has five parameters it will be a first commode of pair will be five parameters gamma it will  be five and then one can write which call it to choose case like this one can prioritize  this monoramy in this way  so a b f gt will be five parameters here so my torso the Laurent polynomials in these  five variables okay so as you get this sink now I should draw so that's  now I can write chain complex for calculating each one of my surface  access coefficient the local system depending on this parameters you know by FGT yeah I  just write you just to see the idea what is going on sorry it was some sick music in  my picture yeah for example one when one calculate boundary of let's see e1 this guy it will be  two minus from this end minus e v2 from another end and so on so you can write what is the  boundary of differential or for example you consider square one this first cell the boundary  will be the following guide will be e5 plus b gt times e1 because to reach e1 I should  go through this things minus e3 plus ie a times e5 it's e3 and to reach a5 I should  go through this monoramy get collection by a yeah so you get this you can write immediately  this chain complex and then first homology will have rank two now I want right in this  complex certain elements when I put here this infinite chain this left-shift symbol and  what is left-shift symbol I recall that we get this one form  can make what form it gives me some kind of this rented relation and what I should do  I should make this will be my left-shift symbols because it will project in under integration  to it's double cover of array symbol it's in a first thing symbol yeah okay so this  is its left-shift symbol it's a left-shift symbol associated some point I it will be  difference of two trajectories as a rented cycles it's one thing is this direction plus  another oriented leaf of relation is opposite orientation it will be kind of symbol plus  vi minus symbol minus vi now I want to yeah but to draw the whole things I should go to  vertically horizontal direction and maybe I just say you I use as a direction theta  it will be zero degrees plus very very small kind of like zero point infinitesimally small  positive direction because it's rational angle I should leave a little bit to rational so  it will be kind of limiting picture so what does it mean I should start to draw from your  text V1 I start to draw things slightly going like this it goes to V5 and then it's continuous  it's returned to itself and then continues itself itself itself so it will be one of  my this because this side is glued to this side if I go from here it's returned I start  to rotate infinitely many times so I get some some kind of like this things yeah it's will  be just one of half and then another thing that should start from vertex V1 but in different  square also continue here I entered E8 so then I continue here from a six I continue  here from seven continue here and so on so it will be kind of another thing and one can  write the formulas for example for this gamma of vertex V1 plus this will be plus and this  will be one will be very simple it will be what it will be h e3 plus a times e3 let's  say square times e3 and so on here because you go around around around what is it I get  one over minus a times e3 and gamma V1 minus it will be more complicated some yeah for  example I consider this e2 I start from here then I reach my six a8 then it should at least  at times next guy e1 plus a square t times e4 plus a cube f times e2 and so on yeah so  things kind of start to repeat after factor to the sort of term so and we make some arithmetic  progression get one minus one a cube f e2 plus a t times one minus f e1 plus a square  t times one minus a cube f e4 yeah so it will be the final sum so you get some rational  expression this plus this so that one can make the second base element get base element  express sources only horizontal lines now I can do vertical things get another basis  and come all to a different base now this change of coordinates change of coordinates  it's still more complicated metrics so as a final formula is like this sorry a bit long  but this is really one final formula and eventually one can make complete computer and get how  do you note my basis gamma sorry gamma Vi okay yeah yeah I can see the vertical directions  will be kind of 90 degrees it will be not exactly 90 degrees but 80 89 999 degrees and  horizontal be zero just like this so it will be certain metrics applied in and the matrix  is the following guy this is the last formula which took me some maybe one minute to write  here we get keys times some another one straws expression  yeah  Just second let's finish  inner glass  Okay, that's it.  Okay, so I get this pretty explicit rational value of matrix value of function.  The rationality is clear because I use some of geometric progressions,  the rational things.  And now one should use some kind of central charge for this monomials.  The central charge of monomials will be the following.  Central charge of monomials A will be plus one.  Central charge of monomials B will be plus E.  And central charge of all the rest will be zero.  Yeah, it's very degenerate, central charge.  And it's responsible for us only for the kind of,  it's not described as a full cross-structure because what I describe,  I describe you go from almost vertical to almost horizontal piece.  Yeah, it will be contribution for this limiting sector.  In fact, one should use eight different sectors.  There'll be infinitesimal goes to vertical and left and right.  One should write actually eight different matrices to get the whole description.  But then the claim, if you start, rotate, change a little bit your central charge.  And then try to decompose in linearity piece.  You see exactly number of this gradient lines on generic run to a generous to surface.  Coming from this monstrous algebraic function.  Yeah, so it's a different story.  Yeah, which one can check numerically.  It's definitely gives the right things.  But I didn't relate yet to this work by Bridgman Smith because I explained that  here I have simple zeros for my billion differential.  But if you come from quadratic, you get double zeros.  And what happens in this double zeros,  it's the following thing in this gradient lines.  You have a pitch critical point will produce not one cycle,  but two linearly independent cycles.  So the matrices you'll have again certain size.  But just not basis, but identification with certain canonical small dimension of  like vanishing commulsion vanishing cycles.  But the whole story is super duper general and works for  complex manifold of any dimensions, if not translated zeros, not compact one.  Yeah, so it's really different story.  But for most the same numbers.  Yeah, okay, thank you.  Sorry, no, this physics terminology, it's some,  there's some lattice and this is some extension of some super algebra and  there'll be some super symmetry.  No, central church, it's a functional on a lattice or  element of some finite dimensional vector space.  And your algebra is just Hamiltonian vector fields and  torus.  Yeah, no, there are two different store either Hamiltonian vector fields and  torus or matrix valued functions on the torus.  Yeah, so it will be like gravity and gauge theory, I would say.  Yeah, because it's in the, in the de-phorophysm and.  What is the type of matrix and this?  It's two by two matrix.  No, any specific like any.  Sorry?  Does it belong to some kind of real algebra or something?  No, of course it's a, it's a,  matrix belongs to two by two matrices.  This coefficients in the what?  Yeah, in fact, it will be series in AB.  In AB, these coefficients in maybe some polynomials in F, J and T.  Somewhere, okay, some such this.  It's all the form of power series.  You expand A, B.  Is it this consequence of the central charges of F, G and  T is zero?  Yeah, it's kind of a very special one form.  Yeah, you make it comes from some homology classes.  FG is built to some homology class and  to get one form, we get this, this numbers.  There is an online question.  Could one insert a spectral parameter in this formula with  the T, with the T's to deform the relation to make it look like a young  backstreet.  I don't know.  It's a three.  Yeah, it's a three.  Yeah, it's a three.  It's different.  It's not two equal to three.  So I have a question.  T in this formula seems a little bit different from the others.  Yes, yes, T, it's just it's kind of AB, AB, FG are really  a model of local systems and T responsible for comparison of  trivialization of two points. So it's a matrix that just appears only  like a very special way.  So taking T equals zero, it's normal.  No, T equal one.  Yeah, so T equal one is the right limit here.  Any other questions?  It doesn't appear in the determinant actually.  Yes, yes, the term to the matrix is one.  You can calculate it.  It's very, just multiply it.  Yeah, in fact, without computer, I wouldn't be able to write this formula.  I have to say it's.  So here you fixed the Chinese two curve, you know, to do this competition.  Do you have a monotony or you change the curve?  Yeah, no, no, you start to change the curve.  You change the central charge.  And then you really decompose this stuff.  Yeah, changing the stability.  Yeah, change stability.  Yeah, here I have no idea what is kind of category theoretic interpretation of this.  Or maybe I have, but I, but I.  Yeah, here it's three spheres, but this is the whole story.  It's in any dimension.  It's not three dimensional yet.  Just just for illustration.  Are there any questions?  It's not the three spheres in this top bundle you are making.  No, no, no, here I think I do do something different.  I go to cotangent bundle to my curve.  I have zero section and graph graph of my one form.  And then for category, I can see the X identify with local and then it's something in this direction.  So I go to Foucaille category of cotangent bundle to this curve, which is spectral curve in Bridgian Smith.  Sorry.  It's time to watch again.  Thank you.
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