 Thank you very much for the interaction and also thank you very much for the invitation for this opportunity Since Professor Casuala's birthday was earlier this year. I cannot say happy birthday today, but I'd like to Offer you happy birthday. Yeah, and the seven tails. So a quantum curve is Still we don't have a good definition yet. So quantum curve is something like It's something like a family of Reese D modules on a curve See and so that We can construct the semi-classical limit. So the notion of semi-classical limit exists on this and this semi-classical limit produces a family of Specter curves in the cotangent bundle of C. So this is a kind of a definition because we don't really Capture everything in this format. So I'd like to give a talk Two parts one is a motivational part. Maybe it is related to the enumeration of Interesting numbers then that is exactly where the physics motivation comes in So physicist a ganagic digraph claim Marino and Waffa wrote a paper around 15 years ago Who are maybe published around 10 years ago where they introduced this quantum curves to capture some Quantum topological invariance of some manifold then later Gukov-Sikovsky and then many other people Are using this quantum curve technique to understand the not invariance. So that's one part of the motivation in physics Yet those motivation Understanding themselves is hard. So let me just start with a mathematical motivation And then the later on I'd like to relate this motivational consideration to another physics conjecture made by Davide Gaiotto in a geometric setting. So that is what I'd like to talk about today. So Mathematical motivation comes from Hurwitz numbers So these are purely a classical object studied by Hurwitz in 1891 and so on so the definition is that you consider a Meromorphic function of a Riemann surface of genus G So let's assume this is a compact Riemann surface of genus G to a Projective line and then a bunch of conditions you impose so that the enumeration becomes non-trivial. So first F has a labeled pose of odors Say mu 1 up to mu N so the number of poses N They are all labeled and these are a positive Integers and then a df equal to 0 has so this is the critical points set of critical points except for the poles has only simple Zeroes And then one more condition because if you impose only these the space of all Meromorphic functions is the finite dimensional manifold So to make it a zero dimension We want to say that the image of these critical points namely the critical values, which are actually branched point In p1 are fixed So then this quantity would become finite Numbers and then well, of course you have to divide by the automorphism and so on But that's a finite number Russian number. I'd like to denote this by group is number of this notation then just Using this representation theoretic Realization of this quantity. There's a combinatorial Equation Well, actually this equation doesn't play any role in my talk So there's no reason to write it up, but since I love formulas Every talk has to have one single formula at least so let me just write down the formula. So first The number of branched points so each branch point except for infinity are simple. So R is a number of simple Ramification points So this is just a solution of a number of solutions of the DF is equal to zero which is actually Easy to compute by Riemann-Hulbitz. So it's a 2g minus 2 plus the end is a number of poles plus a sum of all mu's which is the degree. This is just a sum of all mu's. So combinatorial equation is that if you multiply R to hgn and then mu as a vector Breaks down in several pieces. So this is equal to first half summation i is not equal to j and then So what I'm doing is that when you realize this as a representation of pi1 of punctured p1 At all points you have transposition your sign and then over the infinity you have a cycle type of mu1 up to mun element of the cyclic group and then you count how many are there and that is this number. So what I'm doing is that one of these Transpositions I bring it to the infinity and then merge them together then two things happen. So this is what this is writing so i plus ij so two cycles in the infinity put together and I remove two cycles so cycle length is less one less and then half summation i runs from 1 to n and then I break now single cycle into two pieces with an merging of the transposition so this would cause the summation like this then alpha times beta and then two pieces again so hg minus 1 and plus 1 alpha beta and then a mu Now mu i is broken into two pieces so everything so if I don't mention anything everything else stays and then Set partition and then number partition so genus g so in this case you are cutting Handle of this image curve I mean source curve and then in this case cutting the neck of the source curve So that it will become disconnected then so in this case hg1 so notation probably is Self-explanatory so I don't go too much into this g2 then j plus 1 and the beta Mu j so this is not a traditional way of writing the equation equation can be called in many different ways It is first a similar expression was discovered by Gordon Jackson so I Just use then a terminology cut and join Equation so this this is the formula now So the number is to be normalized in a certain way like like that the mass divide by the Define the who gets number you have to take into account the automorphism, right? So this is auto automorphism normalized. Yes already. So they are not integers. They are just a rational numbers Right and then with this normalization this formula holds So this is an equation for the rational numbers here So the theorem so maybe let me just erase this part Sorry Yeah, it can be zero so this is all possible partitions the all possible Set partitions including the empty set so G has to be greater than or equal to zero and has to be a greater than or equal to one These are the only conditions Okay, so this is what I said equation. I'm not talking about the kind of formula So G and N are peers of also here But of course they are a mu the size of mu is reduced on the right-hand side so from the computational point of view Hand calculation doesn't work Only computer program works because you can reduce the size of mu in this way But that's not my point. The point is the next stage stage so, let me just phrase a theorem and Actually published in three different papers with nijian jang and then also with Beltrand enough and then a broad soft nook and then also Also a Pioto Sukowski says the following number one You then introduce the Laplace transform of this number so define Fgn as a Hormofic function so n Hormofic complex variables to be just a Laplace transform so mu is Just an integer vector of size n and then a hg N of mu then I just multiply irons from one to n of Xi to the power mu i what I said the Laplace transform which means that I'm thinking x variable is simply e to the minus w I Don't use this coordinate. So I just suppressed but this is a Discrete Laplace transform then several things happens then Laplace transform of this equation cut and join equation is a PDE Recursion or induction formula With respect to the quantity 2g minus 2 plus n So the question was that this doesn't allow you to compute a function hg and as a function of integer variable Mu and but here if you do the Laplace transform then you can in actually Change the Laplace transform this equation, which is a PDE but the PDE becomes a genuine recursion With respect to 2g minus 2 plus n. I'm not going to write the formula. This is just a little bit too complicated But nature is exactly the same and then in this formula you do not allow the same gene and then appears So that formula becomes in the PDE So this one is indeed originally conjectured by Marcos Marinho So what is the Laplace transform function of what? What are the space where this this acts ah, so you yes, it's a good question because The question is that does this converge and then what kind of space are you talking about? Yes, you are looking at indeed the simple examples. This is Not the formal passes actually it is a convergent power series and then I Get the box to your and the question to answer later little later at this moment. We are simply looking at that as a Power series. Yes, there's a radius of convergence, which is non-zero right So at this moment just a XR complex parameters So It's a generating function, of course people don't go into the past also But the reason why I want to say Laplace transform is that if you have a question Generating series. Yeah, but the Laplace transform can transform the equation itself to a creation, right? So that's what I meant. So Marinho and then his group So that so this is a recursion existence of recursion was conjectured by him and then a second thing is That yeah, so this is the after long calculations you realize that let's introduce a new variable T Which is again related to your question. So so T actually is indeed a Parameter of a projective line. So indeed Where it is defined is an end product of P1 that is answered your question So let's introduce a new variable T in this Mechanism so let's Let's T be as above Then so I can just plug all these Variables X of t1 dot x of tn. So let me just abbreviate Later. These are just T's Then is a sorry is a polynomial in T of degree 6g minus 6 plus 3n So so to me that that was a very surprising thing and then Marinho did not conjecture this part either. So this is Somehow what happens to be so it is a polynomial in T and then at this polynomial When you write down the formula, it's obvious in some sense. So this is alpha 1 plus alpha n all these are non-negative integers Less than 3g minus 3 plus n and then this is actually tensor polynomial So each variable has one polynomial here irons from 1 to n and then a d alpha i Apply to c0 of t i so I have to and then I'll have to fill in this Rational coefficient. So here D is a differential operator d dx In terms of t I'm using because I'm applying it to t So this is t t squared t minus 1 d dt is a differential operator Yeah, actually Zagie told me that he knew this kind of thing said 17 years ago and so on C 0 of t is actually d squared applied to f 0 1 t So this is a generating function of the Hulu with numbers of the simplest situation namely Rational curve covering rational curves with only one pole So that number you can easily calculate so you can determine this function easily you apply this and then you realize that this function is actually t minus 1 So this is a degree to alpha i plus 1 polynomial and you Just multiply them together then here the rational number is an intersection number So tau alpha 1 tau alpha n times capital omega g n and then this is the intersection over the motorized space m g n bar so here omega g n in General is what? Maximum the mining code Co-homological field theory so usually you have to choose an Frobenius algebra a and then this n tensor Represent this number n here going to the co-homology ring of mg and bar the stable curves of Rational coefficients, but in our case I don't have to really explain what it is because in our case a is simply one-dimensional Vector space so this is a trivial Frobenius algebra field itself and then how it is defined So in our case this is indeed so since it's one dimension just a value is important. This is actually a Total churn polynomial of what is called a hodge bundle Which actually doesn't depend on n at t is equal to minus 1 Where you do have to introduce this projection? Which well again, let me just write in this way The well that doesn't matter I set it to minus one, but of course this t and that here are different things Yeah, give me some letters, please so mg n plus 1 goes down to mg n and So this is the projection Which actually forgets the last marked point here Since this is a universal curve. You do have a relative canonical Line bundle, so this is what I write and maybe a plus one is It's easier to understand, but it's okay So this is just a cotangent bundle along the fiber and then you push it down that is Egn, which is just a push forward g n star of this canonical So this is a rank g bundle differential forms Behave very well in the degeneration of curves on this a dream of a compact occasion So this is not really only they are not only the coherent shift It is actually vector bundle rank never changes anywhere. How you degenerate because the generation is restricted here So this is a vector bundle and then you compute its total churn class and then take t equal to minus one then this is a Conceivich Maninco homological field theory which shows up here And then this polynomial is equal to that polynomial after substitution So this is a part of the theorem we proved This is a modular stack of curves, yes And do you have to work only in the range where usually because sometimes there are many Automorphism, so yeah, yeah stable curves. Yes, it's a modular stable curves. So so it's a obi-fold So neither G is larger because for G was even one. Oh Thank you. I completely forgot to mention Yes, if 2g minus 2 plus n is greater than 0. Thank you and then indeed F01 and F02 are not polynomials Of course from this formula, it's obvious After application of this differential operator, it is a polynomial. So it cannot be a polynomial So only after c0 it becomes polynomial and then these are the coordinate Maximiliana using in your formulation tau alpha Okay, so I'm There is a canonical section from here to here So the data here is a curve with n marked points and then a fiber is a universal curve itself So the curve itself is on the fiber which but in particular has a marked point so you can put this data to that point on the fiber that is a Section here then you pull back this Relative canonical bundle by this section you get the line bundle here and Then the first turn class of this line bundle to the power alpha one is tau alpha one And so on so there are n different line bundles by this section and then pull back take a power So this is the Witten's notation in the physics language it is Correlation function so this bracket is of course integration over this mg and bar right of these Comoraji classes Okay, so one this is just a part of that theorem I see How strong should I push? Okay, is it okay? I thought is a pull down, but you can also push up Yeah so Pull back and the push down both can be done Which one is a vector bundle, okay, so Now let's define Sorry, um, do I want to say anything else here? I don't think so. Okay, so just define c of x and h bar Now x is the original x not the t variable Is equal to just the exponential of a gigantic summation or g or n and Then one of our n factorial because I symmetrize everything and then h bar 2g minus 2 plus n Not h so n of f g n So here I do allow Unstable regions g equal to 0 and equal to 1 and 2 and then I put all x to be the same So let's define this as a wave function So if you don't set so this is indeed the principal specialization of a symmetric function If you don't symmetrize then the exact same expression with the infinitely many variables is a kp tau function So that has been known to Russian school 20 years ago, but now I'm using this principal specialization and then find a simple expression then This solves the following ordinary differential equation, which is h bar x d dx minus x times e to the power h h bar Sorry x d dx. I said the differential equation. It is actually diff differential difference equation Psi xh bar is equal to 0 So what I'd like to talk about I mean I say is that this is an example of a quantum curve It is a vial quantization of some sort of algebraic or analytic curve here So this psi is a solution So this is indeed a rather simple to prove in this particular case And then so what I mentioned is that not theory people are doing is that here you bring colored Jones polynomial You have a difference operator that kills and then this difference operator is a quantization of the a polynomial Things like that have been speculated It's never been proved in those cases. So here semi-classical limit here. What is h bar? I'm going to give you the definition of h bar later in geometry Okay, so here h bar is a parameter Is it real Complex and then here yeah important point is that this expression never converges with the specter h bar And then also it is really strange because if g equal to 0 n equal to 1 h power is minus 1 And then the rest is positive and then I'm still taking exponential So how do you define this kind of mixture of exponential of negative powers and positive powers? So it's all legitimate question at this moment is a formal expression and I'll make sense immediately out of it Okay, so at this moment. It's a h bar is an Parameter where it is not expected for any convergence anyway later. It's a WKB expansion So here semi-classical limit makes sense. So how you do that the question was This one starts with a negative power of h bar and it goes to zero positive and so on so how you make sense Well, I separate out negative power from here. It's exponential. So it's a multiplication and then Sandwich with a negative power there. So what I want to do is to compute negative one of h bar f 0 1 is the H bar inverse proportional term and then I do the same operation You may wonder why I kept x because x you can cancel out but x times a d dx is a Important quantity. So I just kept in that way. So f 0 1 so f 0 1 is a function in x and I make this conjugation and then compute So if you do that So after that you can take limit h bar goes to zero. So the honest computation So this is exactly what we learned in the quantum mechanics first year course Then the result is that so this change into y and then this change into e to the y is equal to zero Where y is definitely here is a d applied to f 0 1 So d is x d dx So this is what happens and then this is the curve now this one can be solved x is equal to y e to the minus y So this is equal to that and then that has a name. It's a it's called the Lambert function And then this appears in the tree counting problem anyway So semi classical limit is indeed the Lambert function, which is Essentially the generating function of zero one invariance. That's what it is happening So these are the theorems we found then So this is an example. So so this curve We call it the spectral curve and then The deformation we'd like to consider here So I said at the family of some sort of d modules So we'd like to consider the family of this kind of object. So here the Deformation we want to do is a following. So we just to make a limit lambda goes to zero of Lambda to the power 6g minus 6 plus 3n and then fgn and then inside you just to scale everything with this parameter Lambda and then so this one, let me call it fgn Tilda which is just a function in original variables. Now, this is the homogeneous polynomial Because I picked up only the top degree terms and everything else is killed So it's a homogeneous polynomial of a degree 68 minus 6 plus n which is indeed having a simple expression The sum of all alphas are equal to now 3g minus 3 plus n and then intersection numbers are exactly what Maxim Considered many years ago and then so it's a tensor product Irons from 1 to n D tilde alpha i c0 t i tilde and then here What I'm doing here is a t very large asymptotics Which means the operator d Change to t tilde as t is large. So this looks like simply t cube d dt and then c tilde zero So t minus one doesn't count so it is equal to t so you plug it in here, then you do get The generating function of intersection numbers exactly what Maxim considered many years ago, and then What I mentioned here is that Laplace transform of cut and join equation Can be restricted to t large and Then this is indeed a virus or a constant condition So this gives a simple simplest proof of the Famous theorem of Maxime's but of course done many years after the discovery of the theorem itself So all that this virus all comes from is this combinatorial formula where you merge transposition into the general type Cycle or Permutation so that's all that was behind the scene is was our understanding of this thing So this gives the motivation one thing I'd like to say is that what did I do here? So in this limit process what happens is that this original omega GN actually change into little omega GN where It is indeed if there were Froben's algebra, then I'm making this to H zero part of mg and bar q So I'm picking up not the co-homology ring, but just a zero dimensional zero-degree co-homology So this gadget is called the two-dimensional Tqft so from the co-homological field theory degenerate into the degree zero part gives you this Computation allows you to make this computation So that is the process I'd like to consider in What is T and what is T i means I didn't understand this fire But T i and T are the same T i and Sorry, yeah, I can say Well, yeah T i is it Better What is the operator then Right, so I applied the operator to the function in T where T is replaced by T i Okay, so each operator you can put T sub i Right, it's a tensor product So you do the same operation and then name these variables in a different way Right, okay, so let me just hurry up And then so this is the motivational part and then the relation between this curve or the Lambert function over there and the differential operator That is the quantization we'd like to study so geometry So this actually comes from One of the one conjecture that's the guy auto made so since most of the things are defined already by Philip earlier this morning So I just rely on that and then now change the gear completely so it doesn't reflect any who's situation now It is a guy auto situation So see is an um ribbon compact women's office It's not really algebraic curve structure. I need I need really analytic structure so compact women's surface and then genius is For the moment greater than two. Well, we would generalize it later Then what are already so let's me use notation G for the complex simple Simply connected again complex regroup but for the Presentation with respect to time. Let me just restrict to the case SL and see then Already it was introduced a double modular space. So this is a modular space of Higgs bundles and then it's important that they are stable ones So I am not considering the modular stuck here the stable ones because I want to change into analysis So he's bundles where he is a vector bundle and the Phi is just a So K is a canonical again. So this is the all linear all module Homomorphism here is a Higgs bundle the stability simply means that if you have any Hormorific vector bundle Which goes the vector bundle F goes to F tensor K then that vector bundle has to have degree purely negative That's the stability condition and then he also mentioned drum visualize space and drum so this is the modular space of Flat connections. So here I forgot to say since I'm talking about SLN So determinant of V is a fixed as a structure shift. So here. This is the Hormorific connection Since it's everywhere Hormorific the degree of the vector bundle has to be zero and then I also Impose the condition stability is a reducible reducible means that the monodromy representation goes to the Zalski open part of the structure group So then I'd like to say the following so non-nabidian hodge correspondence that Phil was talking about Philip was talking about this morning was an a different morphism from Dorbo To a drum and then how it goes is the following so you consider the data Hormorific data from the Dorbo modular space, so it's a Hormorific object then this the stability condition is The system of partial differential equations that has been studied in the 80s by Donson yaw Urenberg those people So if you do the same thing here the stability condition becomes the following so you first Look at the topological part of the vector bundle since determinant is trivial. This is actually topologically trivial bundle So you introduce Hermitian Fibrametric H and then with respect to Hermitian Fibrametric H the complex structure E is encoded into the Chan connection D and then you have this Hormorific phi so you just Yeah, I'm sorry. I'm mixing two things together. So here Chan connection D and then a phi is itself here which satisfies the following non-linear equation the curvature plus Phi and then Phi Hermitian conjugate with respect to H is equal to zero and then the Phi is Indeed the Hormorific. So this is the system of equations called Hitching equations and then at this one is Indeed the the flatness of the following. So D is zeta. This is called the twister line So this is a Chan connection which is a unitary connection plus one over zeta phi plus zeta phi dagger is flat for all zeta inside the C cross so that is the way of encoding the Arzbeck definition of stability into the differential geometric Condition of PDE. So that is what is known. So this and then from here What you do is a solution to so well, maybe this solution HD Phi with a flat D of zeta So that is the equivalent information and then from here What you want to do is that first you look at the zeta equal to one So zeta equal to zero picks up the information of Phi. Phi is a Hormorific one So zeta equal to zero goes back to the original because Chan connection and cause the complex structure of E and then now zeta equal to one what I'd like to do is that the zero one part On the topologically trivial bundle that gives you a new complex structure and then since I define the Hormoric structure by zero one part and the One zero part is automatically Hormorific so that that is the point on the Torbo Mojirai, I'm sorry Duran Mojirai and then this is indeed and the isomorphism is approved by the small-run cases Dornson and Hitchin and then Simpson and then Coret did all the general situations of a high dimensional base so this is what Non-Abelian Hodge correspondences then what Davide Gaiotto said is that let's look at the following family You introduce Real Parameter well, it doesn't have to be positive, but let's me just take positive real parameter And then define this as the same Chan connection plus R over zeta phi plus R times zeta phi dollar So R appears not in the same way as zeta Just R is a risk rating of Phi and then a Hermitian conjugation doesn't Change R at all. So so this is a definition and then now that I ought to say that the limit the scaling limit zeta goes to zero R goes to zero and then this Ratio is fixed So the question what was H bar here you fix the ratio to be a complex number and then this look at this quantity exists So he said that this exists and is an Opah so to be precise this is a H bar Parameterized the family of opas. What is an opa? So opa is a particular point on the drum modular space. So E nabla, which is a connection is an opa Means that there is so three conditions. There is a filtration zero is equal to f n goes to f and minus one Sorry, it's not the exact sequence F1 f0 is equal to e and then two Griffith transversality So the connection restricted to each piece if I goes to f I minus one tensor kc And then the third condition is a real dimension cutting condition so that It generates the grading Gradation so if my which is which plus one goes to f I minus one over f I tensor kc is a OC linear isomorphism. So when you have filtered vector bundle with the connection satisfying this it is called an opa so that's what the Gaiotto said and then this one should be an opa and then so this so what I really would like to do is Try to relate to this with this with a Condition but of course here His conjecture so the so this is just the definition of the procedure Actually conjecture is that if you choose E nabla Sorry, not nabla Let me just write more precisely e0 phi Q is on the hitching section then this limit the double scaling limit Scaling limit is an opa So it is never true For for arbitrary e and phi. Yes. Yeah So if you suppose fix the R and then take the residue of the d zeta R then then You will it is not evident that age bar is fixed and suppose for a moment to fix the R Then take the residue at zeta equal to 0 of this This d d come in a d of zeta R. Then you will find that R is the residue That's so I don't understand why age bar fixed then it would be Well, this is a particular limit. I'm talking about I want both zeta and R go to zero simultaneously Hitching equations it's for R one equal to one. It's not a flat connection. So I have to change the hitching equation here The where did I write? Here R square comes in here Right, right So this equation is modified and this one isn't yes, so to fundamentally we are solving that equation and then therefore Argos to zero would allow would actually force the metric to explode That's right and then so what happens is that maybe this is what I want to say in the end But I may not have time. So you look at this those drum Mosulite hitching section. I'm going to explain in a minute Would that go to by no I'm feeling hodge the real slice of the drum Mosulite SL and R? Connections but then as soon as you change the complex structure from the original a minute later It will be all the same as the drum complex action So it is hypercalous So there's a p1 parameter family of complex structures about everything else is the same as here except for this one so You just change it and then I bring it back Zeta goes to zero as close as original Rescale it then it becomes tangent to the Open Mosulite so that's what is happening H bar is anything H bar gives you actually the opa as a deline connection H by is equal to lambda of the dreams connection So it's a family of Lambda connections Yeah, yeah, so so I have to I want to say a lot of things here So let me just say first I have to define hitching section. Okay, and then yeah come back to your point So hitching section. Yes, hitching section here. I do have to use the constants principle TDS So for the quantization so we call this a procedure quantization completely analogous to the Who is and so on has to depend on the hidden existence of SL 2 So SL 2 is indeed essential for the quantization here. So constants are principle TDS I don't have the time to define but the constant was intrigued by the Poincare polynomial of Simple simply connected the group and then he wanted to Explain the exponent of that factor and in terms of representation theoretic way and then discover So this is the for any such regroup or The algebra of that group they exist up to conjugation a unique three-dimensional TDS Satisfying such a such condition for SL take in case. It's obvious because there's a unique and dimensional representation of SL 2 Any question over there? Oh What is TDS three dimensional simple sub algebra SL 2 principle SL 2 So every Simple I mean a simple V algebra has a I see Well, sorry, I'm just a coping from constants paper So using that so in this case a constant previous It's generated by X plus X minus and an H and then here SLN case is easy to write down X minus is so let me just use the representation This appears in constants paper. So I just a copy down from there So n minus one. Yeah, I do have to stop probably negative two minutes later or something right So and then our R I is I times n minus I so this is X plus and then it happens to be that SLN case it everything is easy so X plus is just a X minus transpose H is X plus X minus Which is just a diagonal matrix of R1 R 2 minus R1 R3 minus R2 up to minus Rn minus 1 So it's diagonal and then a hitching section means that the E0 is Half canonical. So I have to choose a spin structure or say the characteristic whatever so this line bundle and Powered with diagonal matrix, which means that just a Kc R1 over 2 plus and so on all these diagonal entries Kc minus Rn minus 1 over 2 so this is the vector bundle and then Phi let me just write the phi of Q to be just a X minus Plus let me just cheat here because of the sake of the time So 2 to n and then Q J X plus to the power J minus 1 The reason is because I do have to introduce positive roots positive simple roots to make this Expression going through arbitrarily Regroup G, but I don't do that. So here SLN case this works and then the Q is Q2 and so on Qn, which is just an element of J runs from 2 to n H0 of C Kc Tensor J. So this is denoted by so I want to denote by B Felt both wrote it H, which is a hitching base. So indeed from hitching base I am giving you a unique hitching Higgs bundle E0 and then phi 2. Now E0 is Very far from stable vector bundle, but this pair is stable. So it corresponds to the It is a point on the modular space stable Modular space and then for this one and you compute this procedure Defining D of zeta R and then are changing the equation hitching equation with R squared here So and then solve it so then the dot of two parameter families flat again, and then you do have Limit so a theorem is that a lot of people here. So theorem Olivia Dometrescu and Laura Fredrickson Gelbius Kido Narciss and then Laugh Matzio and then myself and the night ski and with myself So we show that the conjecture is true So what is happening? So the question is how do you? Find the limit so quickly how you proceed of this kind of limit First you just choose q equal to 0 and then I ease 0 and then just x minus This is exactly what the hitching considered in his 87 paper So this one is also stable Higgs pair and then you plug into the hitching equation So this actually gives you Harmonicity condition For the fiber metric H, but please look easy. Oh is a direct sum of Canonical to the powers so canonical has a canonical metric if you fix the reman's Surface structure so remain a metric G on the surface namely conformal class I just write it as a lambda squared dz dz bar and they're using this lambda So lambda is a metric of Kc minus half So lambda to the power appears and the fiber metric becomes a just diagonal and then you plug it into this hitching equation And then there are no cues are involved there. So you realize that the solution is Constant-covered geometric So this is a conformal class. I chose has to be fixed as a Constant-covered geometric therefore curve C is realized as a quotient of the upper half plane by the pi 1 of C and then that also allows us to introduce unique Projective structure Projective structure on C coming comes from the solution of the hitching equation then what happens is that around that Metric and then that was obtained by assuming q equal to zero now q is not equal to zero you just that use the Banach analysis to solve this equation, but then a limit is very easy So the formula another formula I want to say is that the zeta goes to zero r goes to zero and then r over zeta is equal to h bar fixed of D Zeta r is actually is a two pieces E h bar is a deformation of easy as a vector bundle and then a nabla q of sorry h bar where H bar is a unique Filtered extension Parameterized by h bar now the definition of h bar here So h1 of C KC so h bar is an element of this so n equal to two case filtered extension is easy I'm talking about sorry. I mean it so KC So I'm talking about the unique extension h bar I'm talking about this one so any element here would give so this is a isomorphic to extension group So this gives you the unique extension and I'm using that so for higher-rank case or general Regroup case you do have to work a little bit, but that's filtered extension idea gives you such a Unique extension and then the nabla h bar q Is as simple as it could be so in this vector bundle Which is no longer the same as before but I'm using this Projective structure you just write down the exterior differentiation minus 1 over h bar and then Phi of q phi of q is that expression that expression itself becomes a connection form In this a new vector bundle. So this is the limit open then a final thing I want to say is that So this one is a joint work with Olivia do me to rescue This correspondence from phi to this final connection does not really depend on that Assumption that the q is a holomorphic section here. You can put any poles over there Each bike or to zero goes back to the original right and then if h bar is not zero immediately All the complex structures are the same so h bike or to one or zero is only the case, but of course you do have a Family and then as you can see this is the doings lambda connection if you multiply h bar everywhere This is the dream's connection. So Since it's time. I just one word Because of this a filtration and then there exists a filtered filtration I mean h bar parameterized filtration this D module here is Principally generated and then you have a single Differential operator globally defined again with respect to this Projective structure and then you apply the original idea of semi classical limit as we did for the Hurwitz case It gives you a global spectral curve, which is indeed characteristic polynomial of this phi So in that sense WKB works semi classical limit to works and then you do have a quantization here. So that is the story. I wanted to say Thank you very much question remarks Just a clarification when you say some limit exists and is an opera in several places. Does it mean that? the Filtration that in the definition of an opera is Is exist I mean is it uniquely defined or is it a part of you have the cost part of the definition Part of the data. No, but when you say the limit exists, how you construct Filtration yes satisfying all these conditions, but you don't claim that it is unique. Yeah with respect to h bar Yes, there's a unique Construction right But he doesn't mean that another filtration cannot satisfy the same property Property not structure, but if you have a if you have a connection, the fact it's an opera is the property of the structure There's a theorem that if such a thing exists is unique Yeah, but the theorem proof of the theorem does contain the actual construction of this filtration satisfying all this right? Sorry, just maybe stupid question So so you started with this discussion was Hurwitz case and then you you you move to this So is so the Hurwitz discussion was the only for motivation or is there some kind of formal relation between? Yes, always that's a question. That's Place we want to go but at this moment we don't have it You see here everything is algebraic and there it was kind of essentially homomorphic system. It was essentially normal So you do want to go to the analytic situation They're all not situations and the grammar of written situations are all over there in the analytic case not in this case There's a question or remarks Okay, so let's sing the speaker again