 Okay, so today is the last lecture, and finally I'll talk about resurgence, which was in the title. Yeah, so what is resurgence? Suppose we get a formal power series in variable, which for us will be like Planck constant, some cn to infinity, and it's called resurgent, it's actually, I don't know because it's really standard definitions, maybe for this lecture is a definition. This is a general idea by Jean-Nacal, what it is, is that resurgence is a following property. We make a Borel transform, it's actually Laplace transform on inverse variables, so we get new series, which is just divided by n factorial, and it's coefficient by n factorial, and the node variable by lambda is, first it should be convergent, radius of convergence is strictly positive. That means actually it's a following, that absolute value of cn is, cn is of n factorial minus constant n, so if we divide by n factorial we get exponential growth, and more. So it's not only just convergent in neighborhood of zero, and admits endless analytic continuation to complex plane with coordinate lambda, which means the following, there exists a countable set, or maybe finite set, most countable set, s and c, which is filtered, s is going to be the active limit for real numbers, s less than l, s less than l are finite. Those guys have growing system of finite sets, eventually it could be everywhere dense, but the limit surface is just c, so it admits continuation to c, just to rely on it. Yeah, I will say this, I will finish, I will finish what's at this end of the continuation, it means the following, that for any, let's see, continuous pass, e from 0, 1, 2, 3, c, such that 0 goes to 0, and c interval goes to c minus s l, s l, maybe continuous smooth pass of length less than l, so if I would find it in many points I'll have analytic continuation of my function, function of this c of lambda analytically continuous long pass, sorry, or b c, it's a coordinate on plane, so it means that c lambda, it means how you denote the coordinate on this plane, I think. So that's, in particular, if you go just by straight pass from 0, by straight intervals, what you get, you get countably many rays, maybe it will be more dense more and more, and you continue to the complement to this rays, and people on computer if you do calculation using pade approximation, it's immediately what you can see from pade approximation, this rays is also on the approximation, but that's it's much more, so it means that you can go around and again see the same picture, and singularities can in principle accumulate. So I'll give a couple of examples, so two examples related to steering formula, so maybe some people already heard it, yeah, so steering asymptotic, if you consider n factorial, it goes to infinity, which is the same as gamma n plus 1, and we know that it's for large n, it has some asymptotic n to power n plus half, and then we get series starting from 1, and we denote this series by sum over c n h n, h bar is, okay, so we get the series. Also people, usually it's kind of not really pleasant coefficients, usually people write its exponent of purely odd form power series, and these are Bernoulli numbers, and then one can make, maybe remove the first, divide by h bar, so start with 1, and denote by sum of c n bar h n, will be, I have another series, which will be sum over k equal to 1 to infinity Bernoulli number, k minus 1, h to power 2k minus 2, so it will be 112th minus 1, 360 h squared, this is essentially same series, but shifted. So, there are two series, c and c tilde, and claims that both c and c tilde are resurgent, and I can write a formula for this Bernoulli transform, for which immediately follows that it's at mid-sandless analytic continuation, bc of lambda is the following, it's 1 over 2.5, 2 pi i, and then I take contour integral, the x, x minus log x minus lambda minus 1, 1.5, the absolute value of x minus 1 is some kind of constant, integrate our small circle, which is less than 1, and less than, let's say square root of, so for small lambda we define by this contour integral, and then this contour can be deformed if you move lambda, so the trouble in this bit values of lambda will be 2 pi i z, it will be discrete set in this case. And for c tilde, the formula is different, Foller plus transform 1 lambda squared over 2 divided by hyperbolic lambda over 2 minus 1, so it's pretty different formulas. In fact, this is kind of more recent one, people this was from for ages, and then by some kind of calculus how we make like product of resurgent function, we want to prove this resurgent by some kind of convolution, and eventually one can try to prove that this guy is a resurgent by form of means, but this was the result by David Servicin, which kind of formally from deduce from this resurgent from this one, but not this nice form, so this is kind of, this is nice form, it's some extra beautiful story here, and yeah, there is still another class of resurgent functions is the following, suppose you get a polynomial in d variables, which is sum over x i square plus high order terms, which will be just some polynomial cubic terms, and then for h bigger than 0, we consider integral of exponent minus u of x over h bar, x 1, x n, and integrate over some disks, like some sum over x, suppose polynomial is some six squares less than some r square for small r, and x i real integrate over small ball, so this will be dominating term, and then it will have expansion sum square root of 2p h bar to power d, multiply by some series in h bar, and this series is again resurgent, and Borrel transform as it's true, it will be a function, it will be solution of p carfuchs equation, kind of Gaussian essentially volumes of the level sets of polynomials, and singularity satisfaction is a set of critical values, or maybe in general sense of p, so that's all what we need about resurgent. Still not isolated, still critical values are finite set, even for non-isolated. I mean that this quadratic part can be 0. No, I don't, what I said, it's quadratic plus high term, it's automatically more point. It doesn't have to be resurgent. Sorry? It doesn't have to be. No, I don't know, let me repeat, okay, p is a polynomial starting a square root. What you said, I'm saying that there is a stronger square root. Which? It doesn't have to be more critical. Yes, of course there are many, many more examples, I just give you one example, just differently. Now return to our story with Foucaille categories and stuff which I talked last time. So recall that from the last lecture we start with algebraic symplectic manifolds, satisfying some condition, we get Lagrangian, algebraic Lagrangian sub-manifold. I will not repeat all business here. So then we get various things, which I didn't spell the last lecture. So we have the lattice of relative second homology, which maps to a boundary map, and also it maps by integral of two form, two complex numbers. And then in this situation we will do the following. We consider some twisted torus and ignore this torus t, which was c star to the rank of h1 of l, and the torus will be, in fact, home from a torus quantizing local system on a, a one-dimensional local system on Lagrange manifold, so home from h1 is the c star. So we get the torus and consider trivial bundle of a germ of puncture disc, it's a long constant, so we have this trivial bundle, and now we make some cuts on cuts in H-plane in small disc, and along the cuts we modify this bundle. We reglued by some automorphism of a fiber nonlinear automorphism, and reglued along certain cuts. This h bar belongs to exponent of e theta alpha r positive. The theta alpha is this collection of arguments of this race. By transformation of the following way, type. By certain transformation, and we want to say what is the holomorphic section of glued bundle. There is my nonlinear problem. This is nonlinear problem. We describe holomorphic sections of reglued bundle, means the following. First of all, we introduce some coordinates t1, whatever, t2 will be coordinates, c star valued coordinates on torus, on the torus. And now to have a section, we should have a holomorphic function when h bar belongs to a sector, a holomorphic function, and if we cross the boundary of it, belong to sector, in fact it extends to some slightly larger sector, and c infinity is 0. So we get a bunch of functions in slightly extended sector, and if we cross the sector, then we get the following transformation rule. tA of h bar goes to tA of h bar, multiplied by 1 plus small corrections, and corrections will be a certain sum over, let's say some classes in this lattice gamma, lying in some strict convex cones, and certain coefficients, depending of alpha, beta, and i, suppose exponent minus z of beta divided by h bar, by t to power of boundary of beta of h bar. So what all this means? This is a monomial, each boundary map gives an element of first command, which is a lattice of characters of torus, so we get some monomial on a torus, and we take product of this ti to various powers, and then multiply by the term, and what I assume that these terms will be exponentially small, so this cone will be sedged, all these things will be very, very, very small, and I approach 0 along the way. Okay, and coefficients grow it most exponentially. What power is t? Sorry? t and the power which is not available. Boundary of beta. Boundary of beta is element of this lattice, and this is lattice of characters. It gives a function, it gives a monomial on a torus. Yeah, so I get such Riemann Hilbert. Did you explain the cone c-alpha? Certain cone c-alpha, yeah. Originally, wall crossing gives infinitely many rays, and I get some kind of cone sitting in hyper planes, and I say that I can multiply all transformations in some sectors and get certain convex cones, give some certain cones. Yeah, yeah, so it's pretty abstract. I think I should give at least one example. Example will be the following. X is c2, this coordinate is x and y, and the form is dx and dy, and Lagrangian submanifold is elliptic curve, punctured elliptic curve, given by x2 is x2 minus 1. Yeah, so it's more a correspond to equations, x of dx squared plus some correction of some functions. You look on such differential equation and you put correction on each bar in a columnaric way. Kind of like this. So, in this case, whatever this relative commulgent lattice, all these lattices are z2, and this is h1 of elliptic curve, and l, which is identically h1 of compactification. Just on punctured elliptic curve. And this period lattice, if you make integral, you get hexagonal lattice. You get hexagonal lattice. And... So, what is the segmental lattice? If you can see the map from z2 by period map to c, this integral... By the period map action... Yes, it's integral, because it's the same as commulgent pair h2. Yeah, it's all these lattices coincide. It's z2. And if you can see the period map from the last model to c, then you get hexagonal lattice. And what is the kind of original wall crossing picture gives you the following diagrams. It gives you six rays. That's 60 degrees. And the transformation, if you go from here to here, you get maybe t1 and t2 goes to t1 times 1 plus exponential minus some constant divided by h bar to t2. And you get this nonlinear map. If you go this way, there's a constant. It can be totally precise here. The constant is... twice the integral of exponent of minus 2 pi over 3 plus 2 pi over 3. It's some real constant. Yeah, so you get this transformation. And all story repeats you have to apply hexagonal symmetry at other rays. So you get a certain Riemann-Hebert problem for this cubic potential. By the way, these transformations are birational maps. And if you apply these birational maps and go back, you get not identity map, you get transmission of order 5. This picture is inside h2. It's inside c. No, this picture is inside ch. By the way, I just want to say that... what about this example? In fact, one doesn't need all these ideas of fukai categories and so on. It's kind of an organizing principle. But in principle, it's kind of this all obvious uproar because if you have equation and put small parameters, you consider stock's data for this. And then in different sectors, you get various... you get control of stock's data to draw exponential things, and ti will be ratio to these exponential small factors, will be numbers of size 1. And then if you go for one description to another, you apply this transformation. Yes, it's kind of... It's end of the day. It comes from compactification of model space, of beta model spaces. Nothing else. Here one has six arrays, but in fact, one can make six cuts on each plane, but in fact, one can make just two cuts. In this case, for example, you have this transformation from all these three arrays, but the composition of this transformation will be a bit more complicated by rational map. It will be convergent along... still it will be convergent along the horizontal cut, where it can just combine the two elements together. So, in this case, you have this transformation from all these three arrays. So, it can be convergent along the horizontal cut, where it can just combine... make it... transformation here, which is compositional, which in fact can make smaller cuts, just... And in general, one can go to... even in a very wild situation, we'll go to something like three cuts at the end of the day. Two is not enough. In general, the image of this map is everywhere dense. You get... three cuts. Oh, maybe before I go on, I just want to say that... for the utilization, I just want to make some... things... not to call it claim, but I didn't really prove it. Let's call it conjecture, that whenever you make such finitely many cuts and such gluing, then... then what happens? And consider any holomorphic section, any solution of Riemann-Hilbert problem for any holomorphic section. The following will hold. So, first of all, T of h of h-bar in each sector, I assume that my function has kind of like C infinity behavior, so it has Taylor expansion at zero. So, it means that it will be sum over T i n h to power n. And this T i n are independent on sector. So, you get form of power series, which does depend on sector because you glue by stuff which has zero Taylor expansion, exponent one of h-bar. So, it's... it's a form of power series at h-bar. And conjecture, this series and as well as logarithm of the series because leading term is non-zero, I still lie in the torus, C star-valued function. This series and logarithm are both resurgent. In fact, this is from some point of view, it's more natural because you don't multiply sector, you have in this variable if you change the basis. So, you should have a bound in the number of weeks. I have this... No, I consider series with coefficients growing exponentially. It's a condition, yeah. The only condition I need is stick cones and exponential bounds. Nothing else. And then conjecture that you get resurgent from... by some kind of... for free. And this will be a pretty special class of resurgent functions, resurgent series because in general definition, you don't say what's the density of the sets and so on. It's a very specific class of functions. I can tell you there's something... You mean for free, I mean you need to have this bound. No, under this condition if I get about LT and so on just let's consider any bunch of numbers with exponential growth. So, what will be this set S? The set S will be image of gamma integral of V, which is also called central charge. It will be finite rank lattice, but if genus of let's say curve L is big, it will be everywhere dense. So the question, how is this filtration all stuff comes into the game and kind of first cuts you get if you try to extend a long race you get only just kind of discrete subset of C, this starting point of this race and containing starting points of race on what people call first sheet of analytic continuation. It will be discrete subset and this will be geometric stations. It will be all Z of which appear in this summation. It will be contained in set of all Z beta. Beta appears in expression. What means if there is first sheet there should be second sheet? Yes, one should kind of really analyze how we can go around how this singularity any idea about the things of the singularity on the further sheet? Yes, not yet. I don't have to really analyze it but my claim that this geometry should guarantee the whole story. So what is the rough idea why it should be true? First one can look on not exactly this question but some toy version. Here we get nonlinear maps, suppose we get some linear maps and it appears in the sum for all these Yeah, in all these expressions again. Let's consider toy and linear finite dimensional model Roman Hilbert problem. Suppose again we get some finite collection of rays and we get interesting we want to modify trivial vector bundle and vector bundle C to powers m again the spec of and we should find some function and h is in the sector and again slightly larger sector the same conditions and when we cross the sector cross the line we should go to certain side i of h plus exponentially small corrections exponent minus certain numbers i j alpha wage bar some constant times mj of h and all these things should be fast decaying along the ray. Yeah, then one can think of these z's and c's as just sort of some constant some values of central, yeah, maybe small z it will be very some complex numbers and just I want to think to be exponentially decaying along the rays and then one can write a basis of holomorphic sections yeah this was written by basis c i j so we get section c i j of h bar j denotes the numbering of this basis element and the coefficients will be delta i j if we know such correction we get just constant section and then add infinite sum sum over k greater than 1 1 over 2 pi square root of minus 1 to power k and here we should get integrals of maybe I need more room maybe I just remove this example so this is i j and yeah you kind of let's see the Newton method of fixed points you substitute cycle to 1 first approximation will be delta constant section but then you substitute here constant section and want to form using residue solution it's itself and so on and then we get certain formula sum of k greater than 1 and you get integral over certain variable which denotes h1 hk and still more complicated formula you take summation of sequence of rays alpha will be slopes and you integrate alpha 0 divided by h h1 hk and you integrate h0 and hi belongs to exponent yeah ideally one should write something like this maybe I take some integral the r is fixed constant so one can do things like this but what can happen alpha can coincide and the sequence one should move a little bit to the contour so that's essentially the formula what you want to get these are numbers I have numbers depending i, j and alpha sorry maybe there are put also c's and put product of c's and arranging r or these things will be kind of easy to understand of the things it gets some solution and from this solution you get kind of by hand certain resurgence because each term individual you get certain absolutely convergent sum and each term has some asymptotic expansion hbar to 0 and this asymptotic expansion will be resurgence by its own means and you get kind of infinite sum of resurgent things it will be actually convergent you get kind of not it's a bit hard way to see that you get analytic continuation and you will not see this you get actually nice monodrome like in other examples j no j it's solutions number j section number j coordinate of c i because the basis have a basis space of solutions and for each number I should get collection of functions so that's how one can do finite dimensional station and then Gaiota Muronetski proposed the following that if you do non-linear changes of coordinates linear transformations of tori produce linear transformations of algebra functions on the torus of kind of c t 1 of algebra of Laramp polynomials which has a basis monomials so it gets kind of infinite round bundle and one can try to repeat the situation and apply the same formulas and it still seems to get exponential growth and convergence that's a rough idea oh my the whole story should I remember correctly that you cannot fix r sorry? you cannot fix just one r as you did in fact for large r you get yeah yeah okay the whole yeah by the way this non-linear all these transformations called this is called four-dimensional wall-crossing because it's related to Vakua in some four-dimensional field theory this is two-dimensional wall-crossing and this kind of like Konsevich-Söbelmann wall-crossing formula and Chikoti-Vaffa wall-crossing formula so it's all related story and then there is a way to kind of really combine this finite-dimensional linear in four-dimensional non-linear things in one package now the story is following all this suppose we have now two Lagrangian sub-manifolds in X then we get two bundles T1 OSCH bar and T2 OSCH we have two bundles and when we modify using these cuts we describe objects in Foucaille categories this will be parameterized object of these modifications along cuts along cuts use complex story we describe objects in Foucaille category panic on H bar when H bar is non-zero very small fibers of modified bundles will be model space of objects now we can consider what are homes between these two objects and home between two objects are roughly the following suppose in one intersection two transversely infinitely many points then for generic H bar homes are essentially given by intersection points or fibers of local system intersection points homes of two objects one and object two correspond to some local system will be sum over I will write C1 intersect L2 but it's kind of fake it's not C it will be more precisely sum of one-dimensional spaces of fibers of local systems homes from fibers of local systems and it's actually not home but Xn n is dimension manifold in the general station for transversely intersect work you get only high X groups there will be no differential low differential here and then when trouble will happen suppose you get two Lagrange manifold L1 and L2 you get some intersection points X1, X2 and the trouble happens if you get a holomorphic disk with boundary on this union so you get certain lattice maybe called gamma 1, 2 which will be H2 X1 you still can get bad trace this is a little bit too rough maybe for each two point one consider homotopic classes of disks you get certain torsos and the whole story will kind of repeat and get roughly speaking you get basis of X groups there will be points which are Xi's intersection point and if you cross the ray then you obtain this big disk then for this Riemann Hilbert problem you get something like this Xi bar maybe I1 and A2 plus maybe exponent minus area of the disk of the disk times Xi2 of H bar and this exponent because it can have infinitely many things it will be maybe you can rate it some kind of constant times product of T to some powers or if it's kind of very big fundamental disk has big area you can rate it's kind of small area plus area of certain powers of T it's a bit long story if it's torsos but roughly you get kind of mixture of two stories not only non-linear change of coordinates but also automorphism of vector bundles and that gives rise and the generation of this conjecture is that all this new series will be again resurgent so there are two types of functions Homes and parameters on modular spaces both of them are resurgent and possible to distinguish whole from the fact no, there's no facts here it's all hopes here so it's very easy and for this game we still a large many faults one can in principle deduce many things for example let's say one is equal to L and X is cotangent bundle so we have this L but for we take cotangent some point Y is parameter Y is parameter so we make this intersection and this X spaces will correspond to intersection points and it will be various solutions and you get intersection points points give some WKB solutions and eventually one get resource of this WKB solutions yeah, you write for example solution equation which has some behavior with some infinity and consider some points, growth, exponentials something and then it gets form power series and that's form power series we'll get again this factorial growth by the same story and or another possibility is the following, you take one is equal to L but the two will be zero section so it means that your interest can in a different question in kind of co homology with K-efficient are home from or to some demodule and just one brief example then Y is C star this coordinate let's say Z let's L1 will be graph of dZ 1 plus 1 over Z 1 minus 1 over Z and L2 is zero section graph of zero form L1 intersected with L2 is just one point and is equal to zero and comparing Foucaille category and and demodules we get comparison between beta and and we get essentially some function of h bar and this function of h bar so I'm trying to intersect Z equal 0 with 0 Z equal 1 sorry you are right you are absolutely right and then you try to and in this case there will be no non-linear change of coordinates because both varieties are genu zero curves first co homology is zero but you still get a linear change of coordinates for these homes and what this whole story reduce to you can see the h bar plane I can see the function let me say right of h bar and real part of h bar b is in zero will be the following gamma 1 plus 1 over h bar and the formula is essentially written here divided by square root of 2 pi 1 over h bar to power 1 h bar plus 1 half you can see this is stuff and you define function in a right left curve plane pi 1 over 4 part of h bar less than zero so you get function one here one here and it will be two cuts vertical line and the function actually extends a little bit so the cut and then if h belongs to higher on one ray we have the following that we have if I'm not confused f right is equal to c left multiplied by and if h belongs to negative ray this will be exponential small term along vertical ray and here it will be exponential small term along negative ray so that's how gamma function appears in this situation I will be very brief here when I explain the example of resurgent functions was a last example was integral of exponent of polynomial divided over h bar and in fact one can try to pursue this idea so we kind of like pass integral interpretation of all these resurgents and why we get and it will give kind of better control of monodromia for this all this barrier transform is the following suppose we have this two manifolds L1 L2 sitting in X and then we consider the following big space space of maps infinity maps say from 0,1 to X 0 belongs to L1 to L2 space of pass connecting all these Lagrange manifolds it's infinite dimensional complex manifold in a certain sense and on this manifold we get a closed one form closed holomorphic one form let's call it nu which is integral of 0,1 of pullback of two form we can relate in families and get one form this over the you push to form integrate over integral get one form on universal cover this pullback of omega of nu will be differential of some function and roughly want to kind of calculate the integral of S over over this on this universal cover and we integrate it over left shot symbols in this infinite dimensional space this left shot symbol started some critical points of S and critical points are 0s of nu and 0s of nu are only finitely many it's just constant map to intersection points so you get exactly this things corresponding to HOMS in Foucaille categories and then if you rotate H bar you start to sometimes change the topology so it looks like you get two-dimensional wall crossing formula with left shot symbols but in fact what is the origin of this non-linear changes of coordinates and the rough idea is the following when we have this fields quantum-mechanic problem it's kind of infinite dimension integral we can twist it the story by rank one local systems here so it's not only one equation but many integrals to calculate it when it is by local systems and local systems for example given by taking local system rank one and one and two and two end points take pullback so you get equation depending on parameterized by product of tutorial and the remarkable story here is that it's in this infinite dimension situation the space of local systems it's kind of became regularized if you change parameters and it became non-linear manifold it will have no abelian group structure that's a very bizarre story here so roughly speaking you can consider gradient lines for this complex valued functional s then you go gradient lines from one point to another and usual kind of story with electric symbols if you change parameters you can break this product of two paths but what is this non-linear change? it says the following if you change the parameter then suddenly your path became extra extra loop this will be kind of sudden change from one path to another and if you interpret this path as a holomorphic disks what happened this one, this holomorphic disk is a map from such guy the usual story is this decompose in one way it will be one problem of generation but these things correspond to this type of degeneration of holomorphic disks when you develop this loop so that some kind of infinite dimension apology which eventually should explain monodromia of this braid transform what I claim is that applying this step-by-step procedure by recurrent formula you don't get really understanding of the right monodromia and it will be a certain understanding at once what was this infinite dimension apology what do you call monodromia? yeah, no, this and this continuation yeah, you get certain representation of fundamental group to infinite matrices with integer coefficients infinite type variation of the code structure yes, yes, yeah yeah, in fact the story is really much more complicated than even this thing whatever in my stories L was always reduced and smooth that could be have an importance this support of Higgs bundle or singularity it could be like this curve example and I claim that in this situation one get new types of asymptotics suppose get singular L and imagine that L is limit of H bar to 0 of some LH bar which is holomorphic family in H bar and all LH are as smooth and reduced so you kind of interpolated by like as per curve by smooth curves and now you do the following you integrate take cycle lattice depending on H bar will be H2 of x LH bar it will be a local system of lattices over C star H bar now have not one lattice but some lattices with non-trivial monodrome and the central and the central charge will be not mapped to complex numbers I'll get Z H bar which will be analog of 1 over H bar Z if L is smooth it will be mapped from gamma H to what you consider piezo a series and consider only negative part but this piezo series form again local system of a billion groups over C star and this will be locally constant map so what I want to say is that if consider volumes of disks you get something like x2 maybe minus pi 4 6 not over 1 you get some exponents A B which belong to the interval from 0 to 1 for this collapsing family of Lagrangian varieties and then the whole story can be repeated but there is no explanation for this through this pass integral story you get completely different or maybe it should be maybe I'm wrong pass integral also can get various powers of H bar in degenerate cases that will be for general family of equations you get much more complicated story and bigger class of gluing questions and for this bigger class it's not reasonable to make Laplace transform at all because it will be you get different exponents so Laplace transform will be not totally adequate tool but still you get some nice class of formal power series which from all this geometry and maybe last kind of remark it's about V store families yeah, suppose we kind of go through all this troubles even through this fractional power story so it means we compactify the family of Foucaille categories at 0 we consider what are all possible limiting objects will be maybe degenerate Higgs bundles and stuff like this now we can apply to the following station suppose we have complex manifold this algebraic symplectic form and to get this class beta 0 in H2 X this class which I usually always have then for large value of parameter R we do the following we consider co-multi-class depending on H bar in the following form it will be R times omega divided by H bar plus I times B 0 plus it's a class in H2 X C 1 2 Y I Z I get a family of parameters for non-commutative spaces and if you look on this think you get real imaginary part real part of the things it's always always symplectic R is some number just for converse maybe can ignore this let's forget about sir? yeah I don't really know examples even I cannot ignore the solution yes yes yes all symplectic if H bar is H bar not equal to 0 infinity yeah so this is and all the formalism kind of works you get family of 4k categories and the property is that 4k category of minus 1 over H bar is Hermitian conjugate 4k category H bar what does it mean Hermitian conjugate it is opposite to complex conjugate categories for categories of complex numbers algebras one can speak about Hermitian conjugacy because this C over minus 1 over H bar is equal to minus C over H because of this property there is symmetry and all this machinery gives a gluing of this family of category to H bar equal to 0 by symmetry we get gluing to H bar equal to infinity and we get kind of twister family of categories and one can speak now about harmonic object as holomorphic sections which satisfies this reality conditions actually I have to live in 2 minutes so we don't have too much time for the question yeah so one can speak about kind of harmonic matrix and in case of we will get exactly about harmonic matrix on demodules whatever it is and if you analyze the number of parameters it looks like it's universal family of this omega n is 0 by interest universally all possible at least in the neighborhood twister families of categories it includes usual hyperkeler geometry as a particular case thank you