 So my talk is about the technique which we invented with Maxime for the purposes of wall-crossing formulas. And today I will explain how it could be related to the subject of resurgence. So I start with motivations in its wall-crossing formulas subject, which was popular like ten years ago. So let me start with an example. Consider an algebra of formal power series of two variables. If you don't like rational numbers, you can replace them by complex numbers. It doesn't matter. And let's handle it with the Poisson bracket. So then for a pair of non-negative integers a and b, satisfying this condition, let's define the automorphism of this algebra b, which preserves the Poisson bracket, which maps a pair x, y in the following expression. Alright, so then the claim which can be checked directly is that these automorphisms satisfy the following formula, which is essentially equivalent to the five-term identity for the logarithm function. Alright, so this is the simplest example of the wall-crossing formula. And in the paper with Maxime in the archive, we proposed a big class of wall-crossing formulas for which this one is just the simplest possible example. Even here we can introduce number k in the Poisson bracket. So this is case k equal 1. And then define a collection of Poisson automorphisms of this algebra and with this bracket. The difference is that you put k here in the power. And these automorphisms, Poisson automorphisms, they satisfy more complicated identities, wall-crossing formulas, where in the left-hand side you can still multiply two basic things, but the right-hand side will be more complicated, sometimes infinite. It will be an infinite product of these automorphisms raised in some integer power. So here product of this raised in integer powers. So these wall-crossing formulas, they were invented for a purpose. So they appear in mathematics in the theory of Donald's and Thomas invariants, say of three-dimensional Calabria categories. For example, Calabria of three-folds. And these integer powers, they are these dT invariants. In physics, the same numbers and essentially for closely related reason appeared in the four-dimensional n equal to gauge theories. And basically this is a development of the last like 10, 15 years at most. But the simplest formula, five-term identity, it appears long before this relatively recent development. It appeared, it can be found in the paper by Dillinger, Dullaber and Famm, 1993. The Varossi surgeons for the operator which appeared in the previous talk and many other talks. It's a Schrodinger operator with a small parameter and VFX is a polynomial. So it seems to be a completely different topic at the first glance. So the question is what is an underlying mathematical structure which naturally appears in both in this wall-crossing story and in the resurgence of the Schrodinger operator. And so that's a structure which we proposed in that paper with Maxim. And that's one which I remind you as a motivation. So underlying structure, the one which we called stability data on gradedly algebras. So by the way it's also underlies these wall-crossings which appear implicitly in Maxim's talk on exponential integrals and I will comment later on that. So what are the data? So I'm a mathematician so I would like to give definitions and state results precisely. So what are the data? So free abelian group. So in physics it's called charge lattice. Gamma graded Lie algebra. Say overrational. So any field of characteristic zero. So in the condition gamma graded means this. Homomorphism of abelian groups. So the notation is chosen in such a way that physicists translate immediately this is the center of charge. And collection of elements. Gamma graded component. Which basically in the sense these are general saint Thomas invariance or BPS invariance up to something which some transformation which I explain later on. So these are the data. So four types of data. And there is just one axiom which we called support property. That's an axiom in this data. So basically this means that if we consider the support of this collection. Set of all gammas for which a of gamma is non-zero. Then this support is separated from the kernel of the central charge. All right. So it can be formulated as an existence of certain quadratic form which is negative on the kernel and positive on the support. So this is an untrivial part which one should always check in concrete examples. So this axiom it allows to define certain, if you'd like certain pronulpotent group. More precisely for every array on the plane. So it's a corollary of this support property. So for every array with vertex in the origin. One can give a meaning of the following expression. So one can take exponent overall of the sum overall a of gamma with the condition that the central charge belongs to L. And moreover from this property one can deduce that these gammas belong to a certain convex cone, C of L which belongs to gamma. So this can be thought as the rays for which this A of L is non-trivial can be treated as kind of analogs of should like subgroups in some big possibly infinite dimensional group. In particular if we take a bigger sector but which has to be strict which means less than 180 degrees it can contain many rays like countably many rays for which A of L is non-trivial and so we can define the corresponding group element which is a product say in the clockwise direction of these simplest inputs. Alright, so this is the axiomatics. On the set of this stability data there is a certain topology which basically means that if we have a continuous path in the space of central charges which is entirely linear algebra data then it can be lifted to the continuous path in the space of all stability data in particular involving this A of gamma which in general they kind of behave in the locally constant way. So in many examples in practice we have not just the latest this latest gamma but we also have an integer-valued skew symmetric form it can be symplectic or not and so if this additional piece of data is given then we can consider the algebra of functions on the corresponding torus this is a torus so you can consider a torus of characters it's a matter of choice. GM, okay let me replace some of these physicists. Okay, it's C star. So this is a linear algebra because this is a... Okay, I can't do all. So again either of these, alright. So anyway, so this is a torus. It's a Poisson torus. So therefore there is a bracket and in fact if we treat it not as an algebra but just as a linear algebra so it's a direct sum of one-dimensional vector spaces so graded components are one-dimensional and so this if you want to... on this let's call it torus linear algebra if we want to endow it with a stability data so this A of gamma will be just numbers. So then the... In return numbers? No, in general no. So I put it C I can of course return to gem so at best I can kind of assume that these are rational but I will tell you in a minute what one should do in order to make them integer. Alright, now returning to this example of Schrodinger operator. So how to see this data? So we have the spectral curve which is hyper elliptic in this case. So it projects 2 to 1 to the projective line and there are ramification points and there is also a singularity at infinity essential singularity. So let's take gamma to be the relative first homology this is the set of all these ramification points and infinity. So there is a Poisson pairing. Inside sigma at this S is ramification points and infinity at all. No, I think it should just come out of open graph and have a really Poisson pairing. Yeah, it will have, so... No. Yeah, it's the first homology. Alright, you are right. But then I need to take the dual one which is the first homology of the open curve and the central charge will be just a period map. Now, in this example so this is gamma there is a natural Poisson pairing and so then the analog of this it's a map T of gamma which maps which maps the generator, inter... So this is an example of Poisson cluster transformation of Fokkengancharov and you can put here instead of minus you can put z2-valued cosine in general. I mean if you want to generalize z2-valued one cosine of sigma. Now, returning to Stavrov's question so at best we can hope that A of gamma are integers, excuse me, areationals but if you have a skew symmetric form then it turns out that the inverse inverse transform of A of gamma tends to be an integer. So if we write A of gamma in this way then... So these are BPS invariance and physics. Alright, so then the wall crossing formulas which are basically just a direct corollary of the definition of the topology on the space of stability data. So it's a certain wall crossing formulas these are identities between T of gamma placed in this integer powers. So then in the example in this five term identity this omega of gamma equal to 1. Now the term wall crossing formula just for those who have never seen it before. So the origin is the following here is a space of central charges it's a vector space and in the space of central charges there are real co-dimension 1 walls where these omega of gamma jumps. So omega of gamma in fact it depends on the central charge and so if you have on one side of the wall if you have some collection of omega of gamma so they are locally constant if a set of walls is not dense subset so when we cross the wall we apply the wall crossing formula and can calculate these numbers on the other side of the wall if we start it somewhere in the space of central charges. Inside vanishes on the wall? No. Okay, thank you. I should say what are these walls so the walls generic walls they are defined in the following way you look in the lattice you look for central charges which do the following there is a lattice of rank 2 like 2 z independent vectors which central charge is mapped to the line so there are lattices which are collapsed so then the central charges are proportional. Alright, now if we return to this example of Schrodinger operator which I use as a motivation for this wall crossing story so in that example we have 5 term identity omega of gamma equal to 0 so of course like Schrodinger operator it's the same as H connection of rank 2 SL2 connection so what we have say SL2 connection curve can be in fact any so then in the Durham picture we have in a fine line the space of connections and the monodramid data Riemann-Hilbert correspondence monodramid converts it into an analytic path Is A of x morphic or amy? A of x can be morphic it can have both so if I use the notation for the generators of this so when I have an analytic path I'm interested only in the germ of this path as H goes to infinity and in the terminology of this 90s so this gamma of H is called Varus symbol wall crossing formulas actually they ensure that you can glue a global solution from local WKB solutions there is also Varus coefficient and the theorem of Akal that these Varus coefficients they are resurgent and resurgent image and moreover Akal introduced the notion which is called direction of singularities which you can easily guess what it is and so these directions of singularities are raised through in minutations through the yellow gamma now Varus coefficients they are transformed slightly differently but it's a direct consequence so I put here plus or minus because I can, as I said I can introduce one cycle so here let me put omega of gamma this is omega of gamma so and Varus symbols they are transformed as cluster coordinates and so then you can interpret if you'd like this observation that in the space of framed local systems which is a cluster variety we have an analytic path near infinity and so then in a sense this appearance of H and all these resurgence it's auxiliary to the fact that there are cluster coordinates in the space of framed local systems okay, now this subject of 90s about Varus resurgence it was it received some kind of boost after this wall crossing story and there are papers devoted to the study of Varus resurgence in kind of more complicated geometries so I can mention a paper by Iwaki Nakanishi a few years ago everything is in archive and even today you see it's non-stopping story even today there is a paper by Hollands and Neitzke exactly on the same topic where they generalized the second order Schrödinger operator story to higher order including Oppers okay, now in this story still omega of gamma this BPS invariance they are equal to one although the wall crossing formulas can be more complicated more than just five term identity but you can since as you see this all about the geometry of the space of local systems so you can ask other questions in particular this now well-known famous paper by Gaiota Murin-Neitzke in fact several papers they wanted to construct hyperkeler metric on the space of local systems it's a different problem but still very interesting it involves the same geometry it involves the geometry of the spectral curve and triangulations with vertices and ramification points so in that story we see more complicated wall crossing formulas with non-trivial omega of gamma not necessarily equal to one in fact the geometric meaning of this omega of gamma is a virtual number of geodesics on the spectral curve with a given homology class so it's another repeatable problem in geometry now so in these examples the Lie algebra is infinite dimensional it's a Lie algebra of functions on the torus so on the Poisson torus handled with natural bracket now in Maxim's talk different type of wall crossing formulas appeared he didn't write them but maybe he did I don't remember so he started exponential integrals say infinite dimension and here some volume form so here the graded Lie algebra is just gln where n is so gamma here if you remember it's middle homology or middle homology of the space so all this in some big space with respect to some boundary at infinity with integer coefficients so this is gln and the gradient is just by the root so gamma belongs to the root latest so the wall crossing formulas here are more simple they are so called so from physics perspective this is a two dimensional massive theory supersymmetric and this is four dimensional but as I explained later you can combine both things still have a geometric meaning now if you have a stability data I erased the definition but there is a central charge and you can always introduce a small parameter by rescaling the central charge so in particular in this matrix integrals excuse me in this exponential integral story you have a collection of critical values of the function S and so then this rescaled central charge is this one so then the walls here the walls which Tibou asked it's the space the central charge basically is given by the collection of complex numbers and so the wall is a collection of complex numbers where three of them belong to the same real line yes it's complex and actually so then you can just having a fixed stability data fixed central charge you can kind of rotate change the argument and so you see some interesting events so okay is it possible to talk about the status I mean there's this sort of conjectural hypercalometric I mean has that been proved now no no it's not proved so the thing is that kind of they have some integral equation that is generalized or dynamical which depends on the additional parameter additional positive parameter some radius have they proved that these functions do give the clutching match for a Twister space or not so you want me I can give a talk on the work of Gaiotomo it's give it will be a different talk but roughly speaking kind of if you start with a Dalbosto if you can start with a wrong complex structure with one which is a heating system so you have a fallation by a billion varieties so you can try to construct first you construct the nave metric which is semi-flat then you add this instanton corrections which basically correspond to you consider kind of geometrically they appear when you have pseudo-halomorphic disk with a boundary on some in this case on the geodesic for the quadratic differential so then you have some integral equation you want to solve it kind of by iteration so the question is whether the solution exists so they prove that the solution exists as a series or a small parameter which is this R which ok so then in a sense it's a non-archimedeon hyper-calor metric but they didn't prove that it converges is that not the only possible sort of geometric interpretation to what these possible swathes actually are though I mean is there a different geometric interpretation? I'm not sure it's a question about geometry it's a question you do iteration so your integral equation the answer is written as an infinite sum over some planar trees and so the question is there a different geometric interpretation apart from that if we did want to understand geometrically what the functions actually are you know I think it's it's just a technical problem of proving some estimates and actually it's a very good kind of a point because I am coming to these estimates which they cannot I mean which are not available in the case of Gaiota Murenetsky it's a special class of stability data generally actually later with Maxim we introduce a notion of Valkrosian structure which serves also complex integrable systems like Hitchin system for which the latest and the Lie algebra they can depend on some base they form a local system but anyway so exponential stability data and this is a class of stability data which as we expect is related to resurgence I will formulate the conjecture later so so then the setup it's graded Lie algebra and assume that it end out with a norm so like every graded component carries a norm it's a Banach norm definition so suppose that this norm graded Lie algebra is end out with a stability data in a sense which unfortunately erased but hopefully you remember so it's central charge a collection of elements A of gamma blah blah blah it is finding the support property so we say that stability data are of exponential type if there exist positive constants such that we have this exponential estimate for BPS invariance for Donald St. Thomas invariance depending on who you are physicist or mathematician alright so this is a definition and so there are several geometrically defined stability data which we expect to be exponential so if you'd like examples so first suppose X is a three-dimensional Calabi Yaw or complex numbers so then the latest is third homology or co-homology it doesn't matter if we assume that it's compact let's assume that so the bilinear form is a Poincare pain and the central charge is the integral or holomorphic volume form should be some Calabi Yaw or three-dimensional cycles so this is one example and omega of gammas are DT invariance so second is complexified Czern Simons it's kind of inching in dimensional type of example but example what are A gamma or omega of gamma better these are called DT invariance like number of special Lagrangians in the given class in three-dimensional like virtual number virtual number of special Lagrangian submanifolds in X even class I think it's even not known whether every class can be realized by special Lagrangian but we are all optimistic so then we hope that this is an example not only every realized but also there is an exponential bound on this number of omega because you can replace in that estimate A of gamma so complexified Czern Simons but actually not probably not halomorphic Czern Simons and the last example complex integrable systems of hitching type so for the discussion how this omega of gamma appears naturally in the framework of complex integrable systems I rather give you a reference to our paper in archive alright so what is gamma in complexified Czern Simons in complexified Czern Simons it is H3 of your three-dimensional real manifolds well it's Z it can be zero I theoretically yeah but it's Z if it's fear is it for halomorphic? not for halomorphic for halomorphic actually I don't know because whether it's of exponential type you have count the number of co-associative sub manifolds I have no idea whether there are results what is A of gamma so given an integer gamma what would be the or omega of gamma that you're counting what will be the omega of gamma what is it? is it number of harmonic bundles harmonic local systems on the three-dimension some kind of custom variance but for complex for complexified yeah yeah but if you want to compute it kind of you can replace any complex representation by I think you can by UN no you cannot yeah you're right yeah it's sort of number improperly defined sense of representations in whatever gl and c where gamma is m where gamma is z gamma is z little gamma here yeah just integer yeah all right so good let me that only does positive integers what about negative sometimes there can be no negative integers yeah for example if you consider representation through quiver your latest is like zn dimension whatever dimensional latest but actually you see only non-negative gammas you see the cone all right now how to approach resurgence so the idea is to encode geometrically exponential stability data as construction of some complex analytic manifold which is in fact an analytification of some formal scheme which exist always without any exponential bound so let me kind of be brief because I don't have too much time so what's the idea so so suppose that you decompose you decompose a plane into the finite union of sectors such that every sector intersect only with the previous one and with the next one no more and then if you have this support property so we can we can construct collection of strict convex cones in this vector space I don't know can be rational and if we cover the whole plane so then we have a wheel of cones this kind of picture all right so then the idea is actually it's very much similar to what appeared in Maxim's talk on exponential integrals when you started with some trivial bundle over each sector and then you modify this trivial bundle with some wall crossing automorphism so it becomes not necessarily non-trivial and the main point was okay so let me do this so in this case it will be locally trivial bundle with the fiber which is a torus so let's consider first a small disk with the coordinate h bar and if we look for the intersection of these two things we have general wall crossing formalism we have automorphisms which I denote by g i and so I have a collection automorphisms g i from the completed algebra of functions on these torus if I choose the basis maybe completed ah okay yeah you are right my all right so gamma is is something which is given which I choose it's part of the data so my l algebra in this case is just a l algebra of say algebraic vector fields on this torus which is graded right so then the map this infinite dimensional group it acts on the basis of monomials and it acts in the following way and if we impose the if we assume that its stability data is exponentials and there is an exponential bound on the coefficients of this series so they define analytic germ and also if you want since I want to formulate a conjecture about resurgence of some series so I can also assume that in each cone I have an estimate that the real part of z over beta over h when h is in the sector and beta is in the cone that the real part is positive so then we can use this if you like that one parameter group to twist these automorphisms in the same way as we went from Vaross symbols to Vaross coefficients in the same procedure so then we have maps for every sector and so then we can use these automorphisms to glue trivial bundles with the fiber key over each sector into something new now because of the exponential behavior exponential decay of these factors exactly the same as in Maxim's talk about exponential integrals the Taylor series do not depend on the sector and so then we get a bundle or the punctured say disc which can be continued to infinity and therefore analytically to the center and so then the conjecture is that this series it's a general conjecture you can check it into examples this series is resurgent and the same is true if you take the logarithm of this okay so I have assuming what this is assuming exponential bound this series can be considered holomorphic sections yeah holomorphic section of the glue bundle but this series they are formal series which are resurgent that's what I mean these cones associated to the sectors or what yes these cones are associated to the sector it's not entirely anything it's almost anything there are some compatibility conditions with the central charge so the central charge is a map from the vector space which contains these cones to R2 so if you consider the dual one you have an embedding of R2 so it should be kind of a slice section which contains which sits inside which kind of I don't know like imagine you have some pizza and you just push it in the center so then the horizontal slice is the image of R2 it's a technical condition which I prefer not to be very technical alright so again there are examples when we expect geometric examples when we expect that these conditions are satisfied so we have resurgence series constructed just from the stability data I didn't forget I simply I look at the time and I see yes there are several and there are just basically two minutes left so many things are expected that if we deform the central charges exponential stability property is stable moreover there are some other very interesting properties which you can define just working with stability data on gradedly algebras for example it was observed both in mathematics and physics like for example Kardov that generating series which appear in wall crossing formula they are algebraic so they satisfy algebraic equations so sometimes you can write down them but we have general proof of this algebraicity based on another property of this stability data which we call algebraic stability data the point is that we have this kind of basic block which we call stability data on gradedly algebra and then we can impose additional properties and from this kind of geometry we can deduce analyticity like in this case algebraicity or maybe some property of the series which used to glue the corresponding global object for example if you want to combine it with this JLN wall crossing with this which appear in these exponential integrals instead of gluing some complex analytic space using these automorphisms where this is just an analytic path similar to Vaross' story you build also a vector bundle over it so then you have a mixture of two types of wall crossing formulas which are known in physics as 2D4D wall crossing formulas it's a mixture of Chikotivap and what we've done so anyway my time is up would it be possible to write similar wall crossing formulas for more complicated polynomial for some structures homogeneous but I don't know how to grade it it's not impossible but I don't know how to do that it's a misleading I mean I was brief because it was in complete analogy with the story about Vaross' coefficients it's in one of the exponential terms in one of the X bar and then Z you can fix you can put Z equal to 1 it doesn't matter it's the same story with Vaross when you write down WKB expansion in the parameter H there is also this parameter on the curve yeah so you can fix it and you still have research but are there coordinate power series in H bar they are expected to be in research in H bar yeah yes yes you can see that Haloborff's section is an expand in H bar get formula power series in H bar okay here it's not an expansion in H bar but the point is that you glue okay let me repeat again so the logic is the same as with exponential integrals you have a collection of sectors on the plane in each of these you have some formal series so you correct them and then you glue some analytic vector bundle but the expansion this is H this is H plane expansion at zero the section is the same it does not depend on the sector so this expansion in H is a resurgence series and how is this related to the standard gram of width and generating functional or donal or kalabijal if you have an exponential estimate for the number of bps states or donals and tomas invariance then you have resurgence you can insert exactly in the way which I just described it's quite general I even didn't use a Poisson structure there for close kalabijal no no no for close it's only for local kalabijal yeah it's only for local for example I don't know conic bundle or spectral curve something like that