 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. There was some topic, in fact, it wasn't really addressed on this conference about wall crossing for three-dimensional Calabio categories in general. So there was a story which we started with Jan quite a long time ago. So the idea is that we have triangulated category and some technical properties we call inconstructible objects form. It can be comprised by algebraic varieties, which is three-dimensional Calabio, and when we have stability structure, then it should produce certain dt invariance, certain integers, level by some indices, and changing stability structure, dt invariance by some certain wall crossing formulas. So it's the whole story and people in algebraic geometry like the following three Calabio categories. Examples of categories from algebraic geometry are the following. You take three-dimensional complex manifold, algebraic variety say, with canonical class trivial. This could be compact or non-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 Gizika stability, and then we count roughly coherence sheets or complexes coherence sheets or 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 something like y is the surface and x is the total space of canonical bundle on the surface. In algebraic geometry it's pretty hard to even construct the stability structures, and there is kind of mirror dual picture. You can see there's some kind of six-dimensional real symplectic infinity symplectic manifold, kind of like dual, because there'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, holomorphic reform. And objects which you consider will be, now consider something called fukai category of your variety, so roughly consider Lagrangian sub-manifolds, but nobody knows exactly what is it. And then stable objects in this picture should be special Lagrangian sub-manifolds. So it means it's Lagrangian sub-manifolds, such as restriction of the three-zero form has this constant argument, which I do not know by theta. Yeah, yeah, here it's a kind of different situation. We don't really know what is the category, but kind of know a priori 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 was some something which we can guess what is going on, namely there was a story constructed by Tom Bridgeland and Ivan Smith. They kind of, they constructed some category with stability structure looking on this picture. It's, they don't directly identify it with fukai category, but it's obviously it comes from this picture, namely, what think they started with? They started with some non-compact algebraic curve and with quadratic differential. The differential, so this small q will be section of curve and square of canonical bundle. It's quadratic differential with simple zeros and they have some numerical condition, so they can have, the curve should be sufficiently, have sufficiently many zeros and the form, quadratic differential should have both infinity. It's expected this should work for compact curve, but it's, 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 two-dimensional variety and it contained spectral curve given by quadratic differential. It will be plus-minus square root of quadratic differential gives us a billion differential defined up to, so two-valued section of these things and it will be smooth curve, spectral curve, it will be kind of doubly ramified with zeros of your quadratic differential and yeah, it's a complex two-dimensional variety and now I construct something like called conic bundle. So the bundle which you 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 with 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 correspond to one-to-one two of the following things. It's called, what's called saddle connections, something very enclosed geodesics. What do I mean? If you have quadratic differential, then you take absolute value to get a flat metric but with conic singularities because it's zero quadratic differential metric is not defined. With conic singularities and conic singularities, the total angle will be three pi instead of two pi. Oh, sorry, no, sorry, the conic singularities you get, yeah, 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 two zeros. How one can proceed, how this one can associate, this saddle connection can associate a sphere sitting in this three-dimensional Calabria. How you do? If you get a 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 disc 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 you get double cover of this thing. You get small interval connecting just two choices of square root which degenerate to a point on the boundary, so you get a disc, you get two-dimensional disc. This boundary, yes, so from these things you get two-dimensional disc in the cotangent bundle, as I did boundary leaves in the spectral curve. Okay, now I have a bundle with this fiber C star and the C star I get the middle circle, so I consider the bundle with fiber middle circle over interior of the disc which degenerate to a point and I get three-dimensional sphere, oh, sorry, what, yeah, yeah, this way you'll 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 cross S2 sitting in this three-dimensional manifold. Yeah, okay, so you got this hypothetical picture and what they really constructed, they constructed categories and these are objects and these will be simple stable objects and construct stability conditions. Yeah, that's complicated work, but, and then by general thing which is also very complicated to get full crossing formulas, but the composition of the thing is really elementary. So what here goes on, you get certain lattice which is H3 of this three-colabia which constructed with integer coefficients, it has intersection form, skew-symmetric form and then you get a map from this lattice to complex numbers. Yeah, there are many ways to see it, for example this lattice can be identified with H2 of cotangent bundle to my curve relative to spectral curve with integer coefficients and then you integrate canonical one form PDQ on cotangent bundle. Yeah, so you get this functional and you get, and this whole crossing formulas, very abstract whole forcing formulas is coming from any three-colabia categories related 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 the answer is 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-emolded, it can move. So you get one parameter family and strictly speaking what will cause on the following, you should consider not only Lagrangian manifold but rank one local systems, maybe this gives you non-recommendant field, so you get something like annulus in some non-recommendant field, this or on Lagrangian manifold, yeah on Lagrangian manifold, yeah so what you form annulus of some non-recommendant field but this cylinder has some kind of endpoints and at endpoints something complicated goes on the 4k category and this thing's kind of like compactified to projective space to P1 and then it gives omega of annulus of closed geodesics will be plus two, the earlier characteristic of P1, yeah, yeah so it's some kind of explanation but one can see that it's just kind of a 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 there's some kind of basic identity, not geometric 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, two, two saddle connections goes to three saddle connections so this is the 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 yeah so it's you get identity corresponding identity with certain elements in certain group which will be simplectomorphism to variables namely you can see that for any integers a and b you consider transformation just in two variables variational transformation which will be cluster 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 zero one you get essentially just just two choices for one one will be one sign one zero 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 another infinite identities for such guys and how we can prove this wall crossing formula directly as we are going to this road to categories and this complicated machinery with whole story yeah so it's completely elementary yeah in fact it was known from many years there was something similar goes on for much simpler group here the group is group collect variational transformation of plane but now consider just three by three matrices unipotent three by three matrices and then there is similar identity namely one one all the rest is zero you get this identity for three by three matrices and if you take a logarithm get elements in some le algebra graded le algebra yeah it's 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 this is the same identity and the question can one see this counting of uh subtle connections uh through this perspective not not through this whole crossing perspective and what the goal of my talk to explain to see yes there is different wall crossing formulas for counting uh uh this almost the same subtle connection I do not count close geodesics now anymore I consider count only uh 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 uh uh but products are and the compositions are essentially the same yeah so uh and the second story uh uh maybe I'll just sell you a few words what are all this wall crossing for what does the wall crossing formalism is in general yeah it starts this uh it's it's almost nothing so we have let's say a lattice of finite rank like z to power z to power n and suppose we have a le algebra let's say overrational numbers which is gamma graded and so the commutator homogeneous element is homogeneous of degrees equal to the sum okay just any le algebra and uh what I call stability structure on g is the following is uh there is e and a collection of a gamma the z is a map from my lattice to complex numbers this additive map so just enough to define on base on the basis of the lattice and a gamma is a collection of elements but for non-zero elements of the lattice so it is saying only one stupid axioms called support property which is the following which is the following there exists the quadratic form gamma cross r uh said that q restricted to the kernel of z z tensor r z tensor r or z r is obtained by extension of scales 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 zero then q of gamma is positive yeah so that's only one axiom and now I think I can remove everything here it's not playing no role no no I I'll uh no it's actually interesting it's maybe related to this thing but uh which it's a good question here no I don't and let's me feel just uh define this abstract notion because le algebra's are very very different nature yeah for example if n equals three it's kind of easy to maybe this left so it's it's I just can see the cube it will be just some part in I don't know in r3 and I project to a plane yeah a square city cannot do it will be my subtle charge and here I get to all integer points or something some kind of lattice is my cube and what what the support property means is the following so there exists certain here it's will be two cones which can can can can uh with the axis is this kernel of my map with this vertical line and set as a whole the all this gammas which it appears with the game are outside of these two cones and when I get a projection I get a discrete subset of r2 automatically because on each domain you get only finitely many things in between so it gets a subset with some kind of polynomially gross growing number in in in a in a disk when I exchange center radius and the order of growth depends on of the rank of your lattice yeah so I get the things and if you go to optimization I would like to think it's like this it's you imagine this maybe subset I know like sphere this too you can see this some domain like this and take cone of it so be very very non convex cone where where is this your elements are present and what is wall crossing formalism now start to rotate a little bit central charge and the claim that you can define canonically new uh wall cross you can uh you are assigned new elements more or less in the same domain and how to do it and in fact one can write in physical form but one can write pretty concrete uh uh things like this so suppose you have central charge let's do it in three dimensional and we project to a plane yeah and divide my uh plane in several sectors and then take pullback on the sectors it means that we divide my things on the sphere we divide by certain kind of like four guns and uh what will happen and 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 which will be the following it will be product increase argument of z gamma uh of gamma such that z of gamma belongs to my sector and I take exponent of let's say a gamma it will be elemented certain pronular important group important important group as such to some convex cone it should take convex hull of this guy uh if it's less than 180 degrees you can include some strict corner get some pronular important group also it's element and some pronular important group and if you subdivide subdivide the things that the element will decompose in the product now what how to recalculate the things if you rotate a little bit central charge it's uh it's it's a fun now now we make a different projection now kind of projection some kind of different way the sphere to our tool and here and here we draw kind of completely different class of uh lines and we take pullback we get something which is not uh we want to take pullbacks of of this race we get kind of hyper uh plane sections which are not parallel to uh don't contain original kernel of central charge but they stay in between and what we go what we do this element uh this element correspond to each original sector I I'll just draw in in uh kind of a locally the 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 the zeta or is by computation sorry are you assuming this hyper plane section is not parallel to the kernel of zeta means that's what you said it's for the different it's kind of new zeta you get zeta prime close to zeta close but not equal and now 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 and decompose by uh two parts and now what you can do you multiply this by this and associated to new sector in for new central charge it will be element and then uniquely decompose in the product of something in a clockwise order yeah so it's kind of define a prime uh whatever for this green guy as a product etc how the uniqueness come every time you're saying unique yeah you can uniquely decompose the element in the because the algebra splits in direct sum of two sub algebras and any element of the same important group can uniquely split in the mean first group and the element of the second group could you could you once again say what was the green and the blue lines green lines will be pulled back of certain finite collection of racing for projection z green lines projection of some kind of different uh uh race 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 now so that's that's roughly of all crossing formalism it's this can be done with kind of like effective computer program if you know in graded algebra how decompose element in elements sub algebra 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 element 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 uh there's a question the general conclusion is that the product doesn't change when you move the race or what's the final conclusion the final conclusion yeah it's kind of locally what globally preserve it's hard to say for example if if you get certain angle sectors for which there's no just in projection uh in in complex in c it projects there's some uh some angle sectors there's nothing goes on this and there are two such sectors there's a total composition will preserve and so what's people called spectrum generator yeah and for dt invariance will be this canonical element for in the case of quivers it happens you can say just one big element on some probably important group stays the same but you decompose in different order but then but what i'll talk about the situation when the projection is this kind of uh the directions of uh arguments of all this projections of these things if you consider race they everywhere dance so there's no one big object is preserved okay online question sorry 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 um when you get central charge with given by rational numbers these boundaries of the course are my rational numbers there's really no good choice and that's a little bit of trouble this you can expect to be some like algebraic transformation uh what else says that in kind of good situations in this hyper planes are rational this transformation uh could have some special property to be algebraic or rational and so on but otherwise uh 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 uh so the lattice is uh for quadratic differentials differential with simple zeros uh so this uh lattice gamma is if i analyze it's maybe first homologer of my spectral curve and we take 19 variant part under a evolution for this yeah this this can be abstract definition of this lattice um i'll get lattice with a scalar product this projection so what is the evolution on a spectral curve it's double cover of your original curve so there's this evolution here and uh but for any so it gets uh uh map from lattice to z and then you get a le algebra g gamma uh and then g gamma will be one dimensional for each gamma and uh this algebra this algebra le algebra will be algebra of function on algebraic torus with some Poisson bracket so roughly it has some i will have some base element and e gamma one bracket bracket with gamma two will be scalar product of gamma one gamma two multiply by e gamma one plus gamma two in fact there are some minus sign like here but i i don't want one can rescale them make it 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 not quadratic differential and not 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 i'm just not my sigma curve if you compare to this situation uh uh on the double cover get one form suppose have a billion differential and uh my group will be h1 of curve zero so alpha with the slightly different group and the central charge is given by integral of alpha so you're considering a billion differential of first type or the any type second time in singularities yeah yeah one can consider actually arbitrary meromorphic a billion yeah and the algebras the algebra will be the algebra will be different yeah roughly speaking it will be metric matrix valued functions on uh algebraic curve on the torus which is again different and matrices of certain size roughly size number of zeroes and what i claim so i get like matrix valued function and this identity will hold like for matrix valued functions on the torus it will be at each point you'll get similar identity when you move 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 m equal one case of curve and suppose you get halomorphic one form which is closed this whole story very general and suppose has isolated zeros and now i'll make something which can when break may continue to assume that zeros are simple zeros the whole story generalized to non-simple zeros uh which doesn't correspond to this story to double cover because when i go to double cover here i get a quadratic differential has something like singularity z d z squared yeah 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 d w on double cover i get double zero not simple zero uh it's additional complication here it's very easy to work out what you're going for for double zeros but i will explain for simple zero so it's not not directly applicable to the previous story yeah so you get this very even high dimensional variety i claim that one get immediately this sum of all crossing formulas uh the claim is the following uh where this whole crossing formulas come from the idea is the following if you have this your variety and one form alpha here now one can let's take real part of one form get real one form closed real one form this will be closed real one form and now one can try to do the following have this real closed real one form you want to represent 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 something function and this function will be function with more singularities of index m uh the m is dimension of my manifold because uh when you get complex more i assume the form has simple zero so it means this locally i can represent complex one form is differential of holomorphic function which will be sum of squares but then of complex variables won't take a real part i get uh sum of m squares minus sum of another squares so if you get something like z one square plus z m square take real part and z i is x i plus y i is the same sum over x i squares minus sum of y squares yeah so you get uh 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 kinematic on x and it gives a gradient flow so what what goes on on on your manifold you get on universal cover get kind of like infinitely many uh uh critical points because consider zeros of my form and go to universal cover get many 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 r m will be unstable manifold uh assert to this more syntax um yeah because they have the same more syntax you 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 pass uh in uh connected to singular points and get set of zeros i consider any pass with maybe the sentence mark points it doesn't belong to positive number that's typical this generic condition because this is the lattice of finite rank and if it's wrong you just multiply alpha by generic uh not not rotate by generic angle and then this countable subset will not go to positive real numbers and then this six is uh this uh this left shift symbols are disjoint there'll be no gradient line kinetic one guy to another at all because along this gradient line uh it's actually derivative real part if r is real part of some complex function and by this condition because it's a 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 uh the integral fun form will be strictly positive number because imaginary part will be zero and real part will be integral positive guy it looks strictly positive so it contradicts this condition so you see there's no 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 of this more snorkel complex that's left shift symbols give homology classes in what in homology of your original manifold with coefficients in some local systems l is a local system of rank one and uh 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 uh so so strictly speaking one should consider l over some non-archimedean field and when you get with the properties that when you transfer uh 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 and h1 are 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 closed chain m-dimensional chain so this element will be classes in hm it will be no other co-homology so in this case it says that for such local system all co-homology will be concentrated in one degree and the space will have certain bases now one can do kind of more or less final um calculation i i want to show you some situation how when uh now what should i do now i maybe introduce some parameter ceta i start to use different uh flows so you get a basis in co-homology or homology say x is local system l local system so class l in h1 of my field x k star and it belongs to certain domain open domain open con direction and uh now for all local cc by some domain i get a basis depending on on ceta because i can start to can rotate and ceta uh 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 x zeros alpha uh i get a basis now i take two elements and i compare two basis elements to two two bases i get transformation from one basis to another and what i get i get a uh matrix valued function on on the torus in size of matrices will be number of zeros and obviously if i divide i'll 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 a different complex structure in one form and uh so the one uh i've done only one calculation in this kind of very simple example uh how to create this change of basis doing from one direction to another and now i'll draw a bit long yeah so yeah 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 uh 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 uh belongs to z plus square root of minus one z and in fact it will be the surface 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 i'll draw you these four squares so surface is glued from this so square some each squares it's in c so this like standard coordinates d z will be this my form will be standard chord z z now should glue the edges yeah in fact i'll have something like still the composition yeah so uh so glue edges so this will be e one glue with this e one it will be first stage from here to here i should identify somehow uh edges the components in the pair how we glue an arbitrary way so e two will be here it will be here t three and now glue it back to three and here e four is glued to itself and now for vertical edges will be something similar e six here will be e six here the seven here will be the same as the seven here so from opposite side and here it will be maybe e eight e 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 bounded vertices you get kind of like v one v one and you get second vertex v two which is okay so you get plenty of vertices now we should write local system let's say trivialized at my zeros i can see the local systems trivialized trivialized at this is my two vertices so it means that for each edge i should write what is the hollow number of my system so that's uh it has five parameters it will be a first common draw pair it will be five parameters gamma it will be z five and and then one can write which color to choose okay something like this one can prioritize this monoramy in this way so a b f g t will be five parameters here so my torsion will run polynomials in these five variables okay so as you get this thing 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 these parameters what do you know by f g t uh yeah i just write your just to see the idea of what is going on sorry so it was something missing in my picture d i think it's v two yeah for example one when one calculates boundary of let's say e one this guy it will be v two minus from this end minus e v two from another end and so on so you can write what is boundary of differential or for example you consider square one this first cell the boundary will be the following guide will be e five plus b g t times e one because to reach e one i should go through these things minus e three was i e a times e five it's e three and to reach a five i should go through this monoramy get collection by a yeah so you get this you can write immediately this uh chain complex and then first commode you'll have rank two now i won't write in this complex certain elements when i put here this infinite chain this left switch symbol and what is left switch symbol i recall that we get this one form yeah when 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 switch symbols because it will project in uh under this integration to it's a double cover over the ray timble timble it's in a per stick timble timble yeah okay so this is its left switch symbol it's a left switch symbol associated to some my point i it will be difference of two trajectories as an oriented cycle 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 thing that should go to vertically horizontal direction and maybe i just say you i i use as a direction theta it will be zero degrees plus very very small kind of like zero point infinitesimally small positive uh direction i i 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 for example from from vertex 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'll be one of my this because this side is glued to this side if i go from here i just 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 six i continue here from seven continue here and so on so it will be kind of in other 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 ah then it should at least at times next guy e1 plus a square t times c4 plus a cube f times e2 and so on yeah so things kind of start to repeat after factor to the third term so and if you make some arithmetic progression you get one minus one a cube f e2 plus a t times one minus u k 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 segment 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 there's a change of coordinates change of coordinates it's still more complicated matrix so the final formula is like this sorry a bit long but this i'll finish it's really one final formula and eventually one can make quickly the computer no get how do you note my basis gamma sorry gamma vi okay yeah you 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 will be zero just like this so it will be certain matrix applied in two and the matrix is the following guy this is the last formula which took me some maybe one minute to write it many glasses here we get t times some another one expression almost finished okay that's it okay so get this pretty explicit rational valued matrix valued function the rationale it's clear because i use some of geometric progressions the rational things and now one use one should use some kind of central charge for these 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 it's not described from the full wall crossing 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 will be infinitesimal goes to vertical on the left right one should write actually eight different matrices to 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 and learn to piece you see exactly number of geodesic this gradient lines on generic rank to a generous to surface coming from this monstrous algebraic function yeah so it's it's it's a different story yeah which one can check numerically it's it's definitely gives the right things but it's i didn't relate yet to this work by bridge this means 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 you have at each critical point will produce not one cycle but two linearly independent cycles yeah so the matrices you'll have again certain certain size yeah but so just not basis but identification with certain canonical small dimension of like vanishing commulsion vanishing cycles yeah but the whole story it's super duper general and works for complex manifold of any dimensions if not translated zeros not compact one yeah so it's it's it's really different story but for most the same numbers yeah okay thank you no this no this physics terminology it's some there's some lattice and there's some extension of some super algebra and there'll be central somewhere there'll be some supersymmetry no central church it's a functional on a lattice or element of some finite dimensional vector space and there's a new algebra's adjust the Hamiltonian vector fields on the torus yeah no there are two two different stories either Hamiltonian vector fields in torus or matrix valued functions on the torus yeah so it'll be like gravity and gauge theory I would say yeah because it's in the far of us and it's two by two metrics sorry now of course it's it's it's a matrix belongs to two by two matrices this coefficients in what yeah in fact it will be series in a b in a b these coefficients in maybe some polynomials in f j and g no somewhere okay some stuff like this it's all the formal power series you expand a a b yeah is it this consequence is the central charges of f g and p is zero yeah it's kind of for the special very special one form yeah you make it comes from some homologous classes f g is built to some homologous class and to get one form we get these these numbers yeah there's an online question could one insert a spectral parameter in this formula with f t with t's to deform it the relation to make it look like a young baxter equation I don't know three to three it's different it's not two equal to three yeah 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 a b a b f g are really a model of local systems and t responsible comparison of parameterization two points so it's a matrix that just appeared only like very special way so taking t equals zero it's normal that no t equal one yeah so t equal one that's a right limit here any other questions it doesn't appear in the determinant actually that's also a question yes yes determinant of this matrix is one you can calculate it it's very you just multiply it yeah in fact a result computer I wouldn't be able to even to write this formula I have to say it's yeah so here you fix the genus two curve you know to do this computation do you have a monotony or you change the curve yeah no no you start to change the curve you change the central charge because and then you really decompose this stuff yeah changing the stability yeah change stability yeah yeah here here have no idea what is kind of category theoretic interpretation of this oh maybe I have but a bit but yeah here it's spheres but this is the whole story it's in any dimension it's not three-dimensional yeah it's just just for illustration are there any questions it's not so these three spheres in this top bundle you are maybe 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 fukai category I consider x identify with local and then it's something in this direction so I go to fukai category of cotangent bundle to this curve which is spectral curve in bridge and smith sorry