 Djartゼly is organizer, produz Enful It is pleasure to be here , partisipateиваетching it and to participate in this very interesting workshop. Okay my title is perhaps just a singular point because the word Richας doesn't appear … We will speak with樹ится ἇ BETH�ᾳ Ἒκmekᾳ ἔχαδᾳ Ἀμεrils ἑ år ៲ Hoo »ះ ἐ ἐ ἅ ἄ​ ៮ Ჷ κα៩γយktᾳ viv ἁ ᵄč ᾳтиᾨ ἐᅡ ៬ ignorance ᵜ Rsq. Იius le �  breathtaking, pretty impressive and good to have tenacity to use, ㄱr is so  funk millions of holes. ɾ�� is the culture in China, ɾ Princess In fact is a Fuchsian. The Fuchsian k, the Fuchsian contribution of the intermediate singularities. Okay so we think that the manner of the composition that we introduce later is a sort of key analog of the pan decomposition in the topology of surface. We hope that it is first steps toward comprehension of this part. Ok, and we hope that I am not a physicist, but I hope that it can be useful in some problems of physics. Ok, so if we look at the qanalogue of the classical differential-Panley equation according to the work of Sakai, everything is perfectly well known on the left side, the side of connection. The left side of the qanalogue of Riemann in Bertmapp. That is the qanalogue of the kamoto space of initial condition. But the right side, that is the qanalogue of character varieties, remain quite mysterious. Anyway, I don't know works about this side. Ok, so it's natural to begin with p6, so today I will present a description of the character variety of qp6. But it's the beginning of the story, so it's for fixed generic parameters. Usually qp6 is complicated for special value of parameters and so on. So it will be a story for later. Today I will look fixed parameter and generically. As I will explain, in the classical case it's a cubic, a thin cubic surface for differential equation. In that case the qanalogue is a modified elliptic surface that I will try to describe explicitly. Some parametrization. And this manual in order to do this parametrization, I will use this manual decomposition. It's a work in progress, not finished, with Yusuke Yama and Jacques Solois. Ok, briefly everybody knows that I think, but it's better to precise the notation and so on. First of all I look at the very classical. My lecture is very elementary sometimes. It's a very beginning for qanalogs. So I look at the very beginning for the classical case. So you start for four puncture sphere. You look at this fundamental group and you look at the representation in SL2C. And of course you know the representation if you know three matrices. So you can identify with an algebraic variety. And then you quotient by equivalence of representation. Ok, and then you get a character variety of the four puncture sphere. I recall it's better to recall briefly the usual computation. So you describe this variety using free coordinate. In Brown you have the fixed parameter. And then you have XYZ. Ok, and then these components, these parameters satisfy the quartic equation. And in Brown you have the fixed parameter. Ok, and then if you fix the Brown parameter, then you get an affine cubic surface and here you have the equation. And so you interpret that as a family of affine cubic surface. And for generic value it's smooth. Ok, so my proposal as I repeat is very elementary. I want to explain what is the q-analog of this surface. I recall that you have a natural Poisson structure on this surface. Unfortunately I don't understand the q-analog. Ok, so I want the q-analog of the character of the variety of the four puncture sphere. Of course there is a natural approach. If you know the fundamental group, the q-fundamental group, you copy the preceding construction. But unfortunately today such a group is not known. So I will use an indirect method. I will start from the classical relation between the character variety and the penalty differential equation p6. And then I will build the q-analog of this thing. Ok, as I explained, it will be the final result which will be an algebraic variety in all the parameters and variables. But today is not the case. Today I will just describe the family. That is the study of the singular case is missing. In particular the study of special solution of q-p6 is missing because it corresponds to non-generic parameters. Ok, here I recall the six p6, the second order linear differential equation. The penalty equation was not discovered by Richard Fuchs. It is well known that you have four complex parameters. Everybody knows that it has a plan-v property, the moving singularities as a pole. And it was discovered by Richard Fuchs in a relation with the problem of isomorphic deformation. And it is that approach that I will copy for q-difference, in the q-difference case. Ok, so it is a particular case of the present situation. Ok, so I consider the modular space of connection with fixed local data, the bound parameters corresponding to the ocamotor space of initial condition. And it is on one side and on the other side of Riemann Hilbert, aim of Riemann Hilbert. You have the character variety F, this one. When this one is smooth, you have an analytic diffeomorphism. Ok, so this is classical things. And then I will try to explain the q-analogs. Ok, so I will not comment a lot on this slide. On the right side you have a fiber bundle with fiber isomorphic to the character variety. And off with a flat air-span connection. And then if you pull back, you get P6. Ok, so this is the basic idea. Ok, now you know the actors and then the play. So the Riemann Hilbert correspondence for Fuchsian q-difference equation was introduced by Birkhoff in a famous paper in 1930. The generalized Riemann problem for linear differential equation and the Hilbert problem for linear difference and q-difference equation. So he had been studied later and will generalize to a regular equation. Ok, so as I explained, I will use as a main tool the work of Jimbo Sakai on q-p6. So they discovered how to obtain a lax pair for q-p6. So they consider a family of function systems. And then you have this matrix with a parameter t. And they put nice condition. And the basic idea is very simple. Extremely, it's a bold step. They consider, they decide that this family is hyzomodromic if the Birkhoff connection matrix is not constant, but is q constant. So if mu from t for qt, then you have this equality. Ok, technically they suppose that the eigenvalues of this one is constant and the other is q constant. This is technical. Ok, and they show that their q is a hydromodromic condition for this family is equivalent to the existence of a lax pair in this sense. The q-analogue, the differential lax pair. Ok, and then this is the basic case, it is q-p6 and later Sakai generalize to other case later Murata generalize to a lot of family of system in particular to irregular systems. Like for classical panlevée, you have p6 which is regular and it's deformation of irregular systems. Ok, so in fact we work on the Murata list but today I will consider only the case of p6 so no resurgence, no irregularity, no irregular equation, no stock coordinate. Ok, then this is the result of Jimbo Sakai and here it's the equation of q-p6. Ok, now so I need some people are used to q-calculus so they are unfortunately not, so I give you the basics. So q is fixed, its value is less than 1 is a choice, it can be bigger but for me it's less than 1. I write sigma q subq is a corresponding operator and I will use the following theta function the formula will be nicer if I put plus but it's like that. Ok, and this one satisfies this very simple, it's the basic irregular equation. So in that sense theta q is the analog of the exponential function and also I need the analog of the operator so it is the analog of the corresponding equation is this one. So you have to, you know when you search for q-analog the situation of differential objects the answer is not unique in general, you have many q-analogs. So Birkhoff in his paper he was not using this one this is the q-analog to x to the power c and there is a more elementary q-analog and Birkhoff uses this x to the power q, this is ramified. So we prefer to 25 years ago I decided to change the Birkhoff approach and to replace ramified function by meromorphic functions. For many reasons it's better. Ok, so now I will describe how Birkhoff built the q-analog of Riemann Hilbert. The idea is very simple, you built a local solution at zero and local solution at infinity and then you compare. I have to, I explained at the beginning that I will describe the things for generic parameters. But to explain some genericity condition in the parameters. Ok, so you look at the spectrum of your two matrices constant matrix and degree mu matrices you look at the spectrum and you impose non q resonance condition. For differential equation you impose in general translation so it's multiplicative. Also there is a condition of functionality at intermediate singularity so this is function at zero at infinity but you need functionality at intermediate singularity and in fact it is this relation and then you have a fixed relation like classical fixed relation that express the fact that you have a five-or-bundle history. Ok, so I will do simple things so I will fix the exponent. I will order the exponent and this will not move. After it will be interesting to move the parameters later. Ok, so the construction of local solution is easy. I look at this constant diagonal matrix I diagonalize this matrix and of course these one are not unique and then they can be replaced this way and then this gamma delta is gauch transformation on the right side and on the left side you have the usual gauch transformation for two different equations but on the character right side the gauch transformation, the gauch freeness will come from these two diagonal matrices. Ok, and then there is an easy limit you construct such matrix with convergent meromorphic, germ of meromorphic matrices and at zero at infinity so at zero is function at infinity is not function but is function up to multiplication by from theta factor. Ok, so as a consequence of this lemma you will get two fundamental matrix solutions of the systems like that so it is exactly like in Birkhoff except that I replace Birkhoff's solution for these matrices but in Birkhoff they are ramified here they are meromorphic. Ok, and Birkhoff use the connection matrix he comes comparing he compares the solution at zero at infinity this connection matrix lives its meromorphic on c star and it is evidently q invariance in fact it lives on the elliptic curve Ok, so you know the Birkhoff matrix in the Birkhoff matrix there is a strong mix of intermediate singularity and singularity at zero at infinity so it is better in some sense to keep as much as possible the singularity at zero at infinity and to in fact replace the Birkhoff matrix by another one that is this one so I skip the contribution to excuse me I skip this contribution this one Ok, so I replace the ellipticity condition by an automorphic condition this one then I look at the property of these matrices Ok, and then m is the meromorphic on c star and ok and it is very important the determinant of m a simple zero on a finite set of q spirals associated to the singularity of the problem and as I explained before you have a gauche freedom then you can change this matrix by this transformation Ok, so you are just led to consider the matrices possessing these three properties the meromorphic over c star the determinant of same zero on the basic singular spirals and then it is the automorphic of course they form an affine variety I do not like that then I look at the action of the gauche group and then I question by the gauche group and my character variety is discussion so I will try to describe it and then you have a fundamental theorem so I associate I start from a system defined by the matrix A of the form shown above on the other side I get a matrix up to a quotient like that so this mapping is onto first remark remark is not injective but the two matrices have the same image if and only if they are rationally equivalent equivalent by the gauche transformation for q connections so I get an isomorphism of set so it is Riemann-Halbert for set ok so now I have to describe the space of monodromy data for qp6 so it is a particular case mu is equal to 2 so quadratic and we have square 2 2 matrices so we name that the Jimbo-Sakai family so it is a particular case of the general condition I described before so you have to consider this space of matrices with this condition you impose and then we impose some conditions like before ok, then we can solve easily this equation it's a particular case and then I have to understand what are these matrix satisfying these conditions or I can replace this automorphic condition to buy these two technically it's better and then I have to study the quotient of this set by the gauche trans the gauche equivalence ok, so I will do explicitly the computation I look at the set of entries of these matrices satisfying these conditions and this is the gauche equivalence ok, in order to describe this question I will use some elementary transcendental functions built with theta functions so I look at this space of special functions that solution of these equations it's easy to check that dimension of this space is k and I get an explicit basis using this function then you remark this one and also it's easy to check that the image of this set by this map is a quadric hypersurface in adequate coordinates so I have a quadric surface ok generically you have an injective map from this set of coordinates of the matrix to this point and the gauche action induce a scalar action like that and these linear forms and they took the value at the singular point these linear forms the great linear forms they span a space dimension 3 so you get a line and then you have this property between this m1 and m2 this part of product of coordinates and then you get a first description of the character variety you look a little bit through that geometrically you get that it is an open affine subsets of a degree to covering of a quadric surface ramifying over a quartic curve the quartic curve is an intersection of a quadric surface and the quadric cylinder in fact it's for four lines close this variety it's kind of algebraic on the same field elliptic curve in fact at the end it is an elliptic curve but for the moment I have only a rough description so I get at the two covering but you don't know what are the elliptic curve at the end it will be a copy of EQ at the moment you don't know so the preceding theorem was for us a state of the art five years ago but at the beginning it was impossible to get a more precise statement and more recently we found a way to get a better description of the character very variety and in order to do that we introduce a new tool that we call Mano decomposition so it was inspired by a paper of Mano a sympathetic behavior around the boundary point of the QP6 equation and its connection problem you will see it's very nice in order to describe this character variety but we hope that it will have other applications because it is a basic thing the main idea is to perform a surgery of the systems using basic hypergeometric systems as elementary bricks so intuitively it corresponds to decomposition of the initial system it corresponds to decomposition of the system in decorated Q-pants and if you look on the right side of the Betty side we will get what we call Mano decomposition ok so I recall the Pant decomposition the Pants are free puncture sphere of course you can decorate the Pants with lace but also with monodromes so this is this picture you have four holes sphere you can cut it in two pairs of pants like that and then you can get two free holes sphere and then you can decorate the boundary with monodromes ok this is the classical picture ok so the basic thing is the work of Jimbo so he explains how to cut the Pant equation into two hypergeometric equations so the proof is based on two different isomorphic formations of Pant decomposition and on the right side you get the usual Pant decomposition I will go back ok so Jimbo moves isomorphically moves these singularities and at the end you get only one singularity so this pair of Pants disappear and you get one side and you do the same on the other side and you get the decomposition and later Mano built a Q-analog of Jimbo work so he decomposed he made a surgery from a cube and the equation he get two basic hypergeometric equations ok so it's difficult Jimbo work is quite difficult and Mano work is quite difficult I will do the easy thing I will look directly on the right side of Riemann Hilbert and on this right side it's a lot simpler ok perhaps I will skip this slide ok so the usual business is that you block you fix local coordinates and you look at these free coordinates but unfortunately we don't know Q-analog of free coordinates so you have to think something different so in fact it's not so difficult we can replace the knowledge of this free coordinate by this representation that is the character variety for the free whole sphere see I will use that as a coordinate of course you have this parameter in some sense of course we need another parameter that is a gluing parameter between the two pairs of pants ok so this is the classical picture so I will explain what is the Q-analog ok so in order to do the Q-analog you have to replace classical hypergeometric functions by the Q-analog that is basic hypergeometric function introduced by Heine at the end of 19th century ok so I will use not the equation but I will translate into systems that I call Heine systems so for the basic bricks of the Q-surgery are basic hypergeometric systems that is at the beginning of my lecture I describe the family of system with a degree mu here mu is one so I look at such system so I suppose exactly I do exactly the same business that in the general picture ok so it's mu equal one case ok so we can apply the preceding description but everybody knows that in the classical case the equation is rigid but here is the same the equation is rigid so the character variety is a point but it's an interesting point technically we can extend the preceding result to the resonant case the character variety is also a point in fact it's written for this case but it's generalised, it can be generalised easily ok and then the idea is that in all cases resonant or not we can consider this point as a function of this T ok and then here is the notion of Mano decomposition I consider these fixed data I suppose that they satisfy some genericity condition I will describe I will take matrix on the right side Birkhoff, the variant of Birkhoff matrix like that then it is possible to cut it in two pieces that is there exist such a matrix in the non-resonant case on this one in the resonant case such that you can write M as a product like that this is the surgery on the right side ok here you have explicit conditions a lot of non-resonant irreducibility and so on such conditions ok it's possible in order to get Mano decomposition you can use Mano result but this result is difficult and anyway it will give you the result only generically but in fact this result if you have this generic condition for the parameters in fact this theorem is true for all the matrices so it's better to prove it directly anyway it's simpler ok you have some evident unicity the only freeness is to the permutation of the coordinate and in the second case it's unique just an idea of the proof is not difficult it's a elementary algebra and using theta function and so on ok so I consider the image on the elliptic curve of the case such that you have this condition then you use this elliptic function you get a ramified covering ramifying the critical values are these four values ok then I consider one of my matrices and then I look at the non-zero column of these matrices of the value of these matrices at the singular point and then you check easily that this one is a projective invariant it's also invariant by Gauss transformations and then you get the image, the xi the manual decomposition you get it using this function you get this set up to the order so in some sense this one is a sort of Q analog of free coordinate ok and afterwards we finish easily there are many cases it's no but it's not difficult ok now I will explain how to use this manual decomposition in order to parametrize the character variety you remember this I have this variety to covering of a quadratic surface but then I will parametrize it using manual decomposition ok so I start from these two values satisfying this condition and then I define hypergeometric matrices like that it seems mysterious but it's coming so well-known connection formula Berns-Mellin Watson for basic hypergeometric equations ok so you have family of matrices parametrize by these guys and they are function of x ok now this is two pieces and then I have to define a gluing parameter so the gluing parameter is very easy such a matrix and then I choose such a matrix c is in c star and then I glue these two pieces p and q by d ok and then it works m satisfies a good automorphic condition and then we can also easily verify that this m is elliptic is an elliptic function of c1 and c2 this is for the generic case and for the exceptional case it's possible to modify to take q logarithm and so on and it's possible to get similar results ok now you can verify with the genericity conditions the determinant the determinant has simple zero on each set of two q spirals and two q spirals and these are only zeros ok and then you check m satisfies the condition so you have you parameterize a big part of the character a small part is missing I will go back later so I will explicitly describe more precisely this parameterization I will use these four points so this map induce an evolution on the elliptic curve the fixed point of this involution of course are these four points so the involution exchange these two coordinates the matrix of the matrix T then you have parameterization you have an algebraic map from the elliptic curve minus four point cross system into the character variable and of course this map is not injective but it is easy to describe what is happening the two points admit the same image if and only if you have this condition so you have an operation of Z2 on this set and then I compute the quotient first it describes the thing I will use four charts I choose the fixed point alpha, beta, gamma, no matter I send it to infinity and so I wrote the classical elliptic curve and then I compute the algebraic quotient so I get an elliptic surface and the elliptic fibers correspond to this value of W this one W is a Yukovsky map ok and then I do that with other charts and we get an elliptic surface ok the subsets are invariant by the action of the quotient so finally my parameterization it starts from this elliptic surface I remove four lines and then I arrive into the character variable and it is injective it is nice parameterization but unfortunately this map is not subjective and what is the missing part it is a union of four curves it corresponds to the exceptional case in Manot's decomposition that is the case when T is not diagonalizable but each curve is easy to prove that each curve is parameterized by C ok now it's not finished the idea is to complete that into a compact elliptic surface then at infinity you add an elliptic curve then you have these four points at infinity and we think that it's possible to find a good solution blowing up at these four points and then you remove some lines and we hope this way to get an algebraic objective map that is really a good description of the character variable and of course you can obtain other parameterization if you cut you know you have four singularities four spiral of singularities I have cut in two pieces but you can choose other cutting here you have the classical analogs and so it's a Q pair of pants it's a pair of pants ok now some it remains some problems and perhaps some idea of applications ok the first problem is the problem of symplectic structure and perhaps somebody could help us here so on the left side the things are well known it's in the thesis of Sakai ok he uses an invariant volume form on c star cos c star like that he spoils on the vertical divisor and in fact it's not proved but we think that your emanusbert is analytic so we get a volume form on the other side but unfortunately analytic and we think that it is rational and it will be nice to describe it directly like Philippe explain you one of the preceding lecture for the differential case ok and then also it's possible not yet done but we have begun it's possible to extend the result to some other case of the Murat Alix list that is species without stocks phenomena but you can extend that to some case of Cupan Bay with stocks phenomena the idea is the same the idea is to build a manual decomposition for such case but you know Qp6 I have split it in two I equations so in general you cannot split it in two I equations because you have to put irregular singularity so for instance this one will be split in Cucumer and I this one will be split in Cucumer and Cucumer and this one will be split in Cucumer plus Annexon unfortunately the method does not work for this Cupan Bay equation in that case we don't know ok so the differential analog here you have a differential analog here you have the usual Pant decomposition but it's possible to extend the decorated Pants decomposition into in the irregular case with stocks phenomena like that for instance Whittaker plus Whittaker that is confluent hypergeometric equations so it's possible to do a Q analog the Q analog is ok so it's Cucumer so people use such decomposers generalize Pant decomposition with nice applications ok there are now some questions to physicists here recently I a student send me his master's thesis more or less a few different equation in 3D supersymmetric Gauss theory so I quote him so he says that it's possible to use to replace for 3D it's possible to replace differential equation by Q difference equation and then he says that in his thesis he explains that it's perhaps possible to use our results on Q difference equations in particular to compute partition functions ok so he says that it's possible to use the basic materials that is built and developed during the last 25 years ok so I hope that Mandodic composition will be useful in this program and Q character variety will be useful in this program and in order to perform that you will need some work in progress by Yohyama and Shangri-Zhang that is all the connection formula for all the variants of basic hypergeometric equations and irregular forms in particular the notion of Mandodic composition seems strongly relate to recent work of Jimbo Nagoya Sakai in the studio of the Q conformal blocks this one I will skip now to end this lecture about generalization and application so it's absolutely immediate but it's easy to extend the Mandodic composition to the case of mu the general mu like that so you cut this one in mu disjoint subsets then by recursively you extend the Mandodic composition the second problem is to try to to extend the Mandodic composition in dimensions strictly greater than n this one is less evident because the equations are more mysterious and so on and so on and so on and the program is to try to do such thing and to try to study relations with Q difference Galois theory and with classification problems and in particular I return to what I explained at the beginning of the lecture who like to build a wild Q fundamental group and as I explain we know the wild component at 0 at infinity and we hope that using Mandodic composition perhaps it will be possible to build the function part of the group so we have spent for 30 years with Jacques Solot we have tried to get this function part it's a sort of non-Abelian class field field it's not so easy we have tried to mimic the differential case the topological case using paths around turning around points but it's impossible apparently but we hope now that using this man or this cut in pieces in rigid pieces it's possible to answer the question and of course another very interesting problem is what happening to this business when Q goes to 1 absolute value of Q first Q goes to 1 of course absolute value but it's a lot more difficult but even the case Q goes to 1 is not so easy if I look only Q goes to 1 then a variant of the Birkhoff matrix is converged toward the matrix which is constant on the slice of the water melon and this allows to recover the usual Mondromy representation when Q goes to 1 you get a differential equation and then this Birkhoff matrix at the limit you will recover the usual Mondromy representation for the differential equation excuse me so the trick is like that so you look at the limit of this Birkhoff matrix and this limit lives on the water melon you have a water melon you have the slice and at the limit the matrix on each slice the matrix the limit is a constant and then the jump from one slice to the other it's a Mondromy okay this is the end in fact I don't think it's so difficult because what you do you even describe with this torsion shifts yeah you can see the coherent shifts with two anti-harmonic infiltrations yes yes they try to bundle so the question for break geometry and then you get to one but the problem is to do explicit computation no it's not difficult it's not difficult but you know it's necessary sometimes it's basic but here it's also got not on just two algebraic structures the same analytic variables infinitely many because they make model invariance and for elliptic differential equation you get like some SL3z but we try various algebraic structures the same algebraic variables yes you get kind of huge family of algebraic structures the same no but you know it's not difficult but what not so easy before getting this model this pancake composition but explicit question of break geometry of vector bundles yes no it's very elementary it's a question of vector bundles and elliptic curves no no it's not difficult but it's necessary to describe such an object it's a basic object and then you get some surfaces and to this volume element it's quite obvious from it's an algebraic simple structure it's completely obvious from ok so you can explain it thank you very much for special values of this parameter it's capital A B C B that you have this cube character variety is independently known to be spherical double affine take algebra that's a theorem of a long term I don't know for special values there is a monodromes here family