 Thank you very much for the invitation to this colloquium. I tried very hard to be here, including climbing over a fence yesterday night in order to sneak into the guesthouse, which was kind of non-trivial. Yeah, so there was some trivial physical effort going on. Yeah, I shouldn't do this. That's wall crossing. That's wall crossing. Absolutely. So what this talk is about, I had a bit of physical limitations where this blackboard tried to do my best. So this talk is going to be essentially a tale of two characters. So two characters. Characters as characters in the play. So the card number one is for a physical point of view, a topological gauge theory. So gauge theory, which in my case will leave on a real three-dimensional, smooth manifold. And for a mathematical point of view, this was proposed by Whitton in order to describe, well, a class of invariants derived from quantum groups. Quantum invariants. Sort of a mathematical counting part of the real. Number two is where, from a physical point of view, so I understand we have a mixed audience today, is, again, a topological theory, topological field theory, but coupled to gravity in two dimensions. So first quantize, topological string theory. The particular version that I will be discussing today is something that is related to care countings on killer manifolds, actually on caveat manifolds in dimension three. So the mathematical counter bar is, well, your favorite care counting theory. These are the two values. This is what they are. I'm going to describe them, and momentarily, but before I tell them, before I tell you who they are, let me tell you what they will do in my talk. So I got two messages. So the first message of this talk is that, well, character one, well, understandable conditions. So both of these guys will give you a bunch of numerical invariants that can be naturally organized in generating function forms, and my claim is that generating functions for the first guy under a suitable identification of characters will match with generating function of care counting invariants of Caldeos. So this is an identification that was put forward in physics literature by Gopakumar and Bafa in, I guess, 98, and further elaborated upon by Guri Bafa in 1999. So this is the first message, and the second message is that both of these guys under essentially the same conditions so that the first line holds our goblin, the normal intent of local recursion, that was introduced this morning by Bertrand. So the plan of the talk is just a very broad overview, is to, well, I assume that not everyone is familiar, at least with one side, either side of the correspondence. So I'm going to tell you, first of all, what I mean, what is sublogical gauge theory, or the spot that's named in the title. Then I'm going to introduce its string theory counterpart, co-conic counterpart. Then I'm going to give you some evidence in favor of the existence of this correspondence. And the evidence in question, so it's good for pedagogical reasons to deal with, once chief example is going to be for the most part a review of the original results of Gopaku, Maruguri and Bafa, but this does not stress just a pedagogical need. The thing is that there is, at the moment, no structural theory behind the existence of this correspondence. What we have is essentially, this is essentially phenomenological theory. There are a bunch of examples where we have some belief, often by physics, that an identity like that should hold. There was a 120 page paper by Wheaton today. Oh, yeah, yeah, yeah, yeah. That's a bit different. That's actually the Gopaku-Maruguri interpretation of the logical string. So yeah, it's not what I'm going to talk about. Yeah, when I started to grow up, I was like, oh my God, I had this little Tuesday. And then I'm going to discuss this, the relation with topological recursion, and since most of my talk will be of expository nature, and on this one example that is well understood, sort of needed and we will have by the end, is to extend the circle of ideas to, well, as much as possible, essentially. We've brought this class of three manifolds, where we believe that such correspondence exists. So there's extensions of the circle of ideas, obstructions that I will try to carefully sweep under a rug, and at least one application to Gramo-Wheaton theory, in particular the Krappon resolution conjecture. So I said nothing in the first five minutes. I guess it all can start. So let me start off with, well, M, a closed, pre-empted, I mean, working too small. Is this readable from the last one? Well, a real free manifold. I'm going to fix G, a compact meager, which throughout most of the story will be C1. I'm going to consider a trivial G bundle. A will denote a smooth G connection. And E, I'm also going to fix kappa, an integer, and R, an irreducible representation of G. Out of this data, I can define a functional, a transgression form of the second chart character, which through a connection A, assigns the integral over M of kappa over 2 pi trace, of U n will be trace in the fundamental representation. Just fix U n for the moment. A YGDA plus 2 3rds H1. So this all depends on the smooth structure now. We haven't picked the remaining matrix in this. And Whippton, 1988, proposed that the topological invariant of three manifolds, depending on all this data, will arise from sort of averaging of the connections. So if you're a physicist, you will consider the path integral over the space of connections module of vertical transformations of exponential I s, which are signs of A. And there are, so there's a number of things that we can compute out of this, the sort of higher dimensional animal god, matrix integrals that Bertrand was introducing in the morning. And there are natural quantities that we can compute. What is the partition function? It's going to depend on our three manifold on our integer kappa called level. And if I stick to the fact that G is equal to U n, it's going to depend on the rank of U n. Or if we have a knot, we have kappa, well, k, a knot inside our three manifold m, we can consider where the homonomy of a gauge connection A and take the trace energy, or E rep, labeled by, by left R. We can normalize this guy by the partition function. So I'm going to denote this by W. This is what a physicist would call a Wilson loop. It's a function of m, kappa, n, and our representation. Which is not the same. It's not the same power as you use to normalize. No, it's the same R. It is the same R. This R is that R. There is no R in normalizations, I guess. No, no, no, normalization. It takes trace R. Yeah. So yeah, this is fixed. This is fixed once and for all, and this goes for the right. Let me call this star level star. So some remarkable facts about two quantities. So the CES in the subscree of the actual sense of what you're assigning is. And this is what a physicist was called, quantum-insured assignments theory, a gauge-graphed unit. So if you're a physicist, there are many expectations that you have and that you can prove at the heuristic level. So the first is that this is a topological theory, which is obviously true from the classical viewpoint, but you can prove that quantum mechanically this is preserved up to minor tweaks in the phase of the partition function and of the Wilson loop. So what this gives you, if you can compute it, is a topological, a smooth and topological three-manifold invariant. So the slight tweak that I didn't discuss is in fact an invariant of frame three-manifolds. And the second and perhaps most remarkable fact about this quantum field theory is that it is exactly so by canonical quantization on the Riemann surface times the line. So by canonical quantization, we can show a remarkable relation with rational conformal field theory. And in particular, the would-be physics invariance would compute this formula turned out to coincide on the nose with the invariance introduced by Grasciotekin and Terayev for frame-notes and frame-tree-manifolds arising from quantum groups. And in the case of knots, you can recover the homelyskeen or Kauffman in the case of which you switch from U.N. to say S.O.N. or S.P.N. So there's a whole class of topological invariance of three-manifolds and of knots that arise this way. But there's another approach to other than using canonical quantization which is to treat a path integral formally as a cubic correction to an otherwise linear and well-defined Gaussian theory. And this makes perfect sense to solve in the limit where the equivalent of h-bar for this QFT is small, which means large-cup. So this can be, the problem can be approached by looking at perturbation theory, the large-cup, this framework. This assumes perturbation theory around a trivial connection, subsume the theory of oscillating invariance. This can be seen, for example, in temporal gauge. But a second type of perturbation theory which was alluded to earlier this morning by Bertrand is to consider instead of the limit of large k, you can consider the limit of large, sorry, this is Kauffman. You can consider the limit of large kappa and N. So again, this is because you stick with U.N. and you get kappa-gram, the ratio of kappa-gram fixed. So as for n-gauge theory, or as for matrix models, in the case of the remission ensemble, when you consider the integrals formally and you have a diagrammatic expansion of the partition function and the vets of gauging invariant operators, there's a, so if you consider the log of star, so the log of z, let me call it f, log of z, m, n, call this f. As a formal expansion of large n of the type sum, g greater than 0, n to the 2 minus 2g at the g of, well, m and the ratio of kappa over n that I'm going to be fixed. And I've just erased them, but if instead of considering gauges of, say, short polynomials of the Carton element to represent your whole anatomy, you consider, well, Newton polynomials. There's an analog of the multi-trace operators, connected multi-trace operators that we're trying to introduce today. W is m and kappa, n, and then I have, well, a string of integers. I'm going to consider a vector of integers h, this is the sort of cumulons. Well, the exact animal got the cumulons that we're trying as considered earlier today. Here I've got a sum 2 minus 2g minus the length of the string of integers times some function wg depending on n, kappa over n, and the integers that define my traces. So this is kind of reminiscent of what was introduced in this morning's talk. And if you look, for example, at just the first line, well, if you're a physicist, you would say, well, if I interpret 1 over n as a string coupling constant, this is, well, the free energy of some theory of first quantized strings, some string theory with some target that will depend on my tree manifold and whose background will be somehow described by this range. And mathematically, there's sort of, this moment is at the stage of just wild speculations. The real question is, well, is there any such thing? Is there a string? Or is this the kind of gadget that would come when you want to pack together invariants that arise from any homological field theories? Just integrals over mg and bar during a string or a co-FT interpretation. This is just no grounding for the moment. Just a belief that we may have is what is actually inspired, toffed in his paper in 74 in the context of QCD. It makes sense to pose the same question here, and do we have a shot at actually giving an answer? So what is h with the arrow? Right, so instead of considering that's some normalize that's a short function of some carton element, u, I want to consider, well, Newton's point, averages of Newton's point. I'm just changing bases in the space of symmetric functions. p, h, e to the u. So I get just powers of, power sums in carton element where the occurrence of each trace is labeled by a vector of integers h, which is zero from some point on. So what do we want here? And I take the connected part. This is vital in order to have one of our expansion that is analytic up to some pole. So this begs for the introduction of some other gadget. So since we started from some topological gauge theory and we want some string theory that is dual to it, it's worth looking at topological string theories. And I'm going to look at one particular version of it, which is a topological A model. So this is the theory that arises when you twist n equals 2 to sigma models on to some scalar manifold, x, it's implanted for omega, and what you're interested in is an intersection theory on a module of stable maps to x. So the fact that the string theories are topological means that they have a purely instantonic character, as you compute observables that are in the homology of the topological B.R.C. operator. So they localize on classical equations of motion, which in this case is Cauchyry. And you want to construct a modular space of holomorphic maps to your given target space. So this modular space of morphisms from a source-projective curve of genus G to x, so in fact I'm going to take this pointed P1, Pn, to x where the image of fundamental class Cg, some class beta in H2OXZ, and I take quotients by biolomorphic equivalents. And as was done earlier today, I'm going to consider a Konciovich competition of these guys. So I'm allowing double points on the source curve and I'm contracting components that have an automorphism group of infinite order. So if we restrict ourselves to the case where x has dimension 3 and it has trivial canonical class, there are sort of two... Well, there is one obvious iterating function of intersection numbers on this modular space. You start with the case where you have no mark points. Find f with the twiddle G sin parameter t to be equal to... This is a sum of two homology classes. In Z, with it by some fugacity, so this is what... So the t-parameter here is dual to the index that goes for the right of all numerically effective classes on x. And here I have, well, the degree of the fundamental class, the virtual fundamental class, this is to connect the logarithm of the partition function of the topological aim model. This is something that has... When you organize this as a sum over the genus, it's something that order by order can be matched with free energies of whatever has one over an expansion. So we got that. Let me call it star second. And as far as the double stars go, we can introduce mark points, double gen twiddle, depending on the same parameters in the second homology of x. We're going to throw in, well, a bunch of parameters into mark points insertions. And, well, simple as that's historic. Let me say this instead. So this is a correlator of... Well, the kind of homology class is that you can consider for intersection theory on this moduli space. We've seen already an instance in the morning. As for the moduli space of stable curves, tautological and bundles that are perfectly defined here as well. You can consider the curvature classes. There are side classes that I can pack in generating function form. And one of the things that we gain from the fact that we have some target space is that we can pull back homology classes from the target by evaluation morphisms. The sort of most general PRC operator you can consider in this theory. Now, when x is a kalabiyao, this contains no extra information with respect to this. There is some universal differential operator in the variable t, out of which you can extract this W tilde, starting from f tilde. But suppose that x admits some holomorphic isometry. Suppose in particular that x is historic, so you have a rank three group of holomorphic isometries that you can play with. Then in this case, the degree axiom of ordinary grammar will continue to break down. And this allows this guy to be actually much more refined than its non-equivariant counterpart. So, if x is historic and you work equivalently with respect to a torus action, this is actually quite a rich object. So, phi-off is here. Phi-off of i, classes in x and up-size. They're first-gen classes of the tautological line bundle that have already appeared in the same definition, essentially, in the case of just stable curves. So, I'm going to take this as a conduit for a double star. What time did it start? 2.10? Half an hour. So, right. So, in particular, when g is equal to 0 and n equals to 1 and double star prime is so-called, is given pause. So, in here, I'm also... This is going to depend on the details of my torus action. This will depend on one integer ambiguity in the choice of torus when it's down. Let me parameterize it with an integer f. This is basically telling me what torus action I'm looking at. So, as a smaller side, if you know about this, since it was mentioned, more or less, in Bertrand's talk, when x is toric, is toric halibut in 3, there's a... well, there's a sort of calculational definition of open-ground within there, by localization. So, in the front of localization, this is q. And if you've heard about this, all the generating functions of, say, counts of holomorphic maps from open-ground surfaces, genus G, and h components of the boundary relative to some is a toric grunge array. These guys are computed in a sort of discrete transform of this double star prime. So, this will depend on, say, for a case of one whole, it will depend on well, homologic parameters T, your choice of torus action F, it will depend on the choice of your Lagrangian, and it will depend on a winding number variable D. These guys are at the topology of a swan times a two-rear plane, and this is a sort of... So, let me introduce a winding number variable associated to the one you go over, this S1. And there's a kernel given by the other beta function of minus Df, Df is D minus 1. Df plus 1 to the minus 1 applied to W tilde G1 of T and F and z set to 1 over D times WD over D squared. So, open ground between there and sorry, we construct it. Well, actually, it's different. What the localization formula tells you is precisely how to recover this open string count from the design... Right. Yeah, great point. So, this guy will depend on a bunch of homological data. If you take the alpha 1, I'll find two parameterized... Well, to index the homology classes that are concentrated at the fixed points of the torus action, then this will compute... So, picking like one particular fixed point will select a Lagrangian such that the holomorphic disks in your open ground-wittern count will attach to that particular... Well, the cap of your homomorphic disk will attach to that particular fixed point. So, they're sort of buried in here in the choice of Lagrangian. So, suppose there is even a very little chance that Witton-Rachateckin derived invariance of a three-manifold M can be reconstructed. Can be given an alternative interpretation in terms of long-wittern theory. Well, the question is what's the target given some three-manifold? So, let me give you the example. So, take M to be simply connected and at this stage I'm not considering either no knots, such as the partition function or the trivial knot. Now, in this case if you take the one over M so let me actually introduce parameters lambda 2 pi i over kappa plus n and t equals 2 lambda times n then as a function of these two parameters f of s3 lambda t on general grounds has some definitely one over M expansion. So, some lambda expansion for positive g with some fg of s3 and t and this guy is computable in closed form as I said. Trincylin's theory is exactly soapable in this case where we have exact formula and we know how to do asymptotics to all orders in one brand. So, in general zero this gives a polylodorhythm of order 3 in the exponential of t plus a quadratic part that I'm discarding at this stage. 1, it gets minus 1 and 12 logarithm 1 minus c to the t plus a constant term that I'm actually discarding at this point and higher g to this it gives essentially the dual characteristic of Mg v2g over 2g, well divided by 2g minus 3 factorial 2g minus 2 factorial times some rational function in the exponential of t which is actually a polylodorhythm of order 3 minus 2g of e to the t plus some constant term v2g v2g minus 2 these are Bernoulli numbers over 2g 2g minus 2 2g minus 2 factorial. This may or may not ring a bell but around the same time where this asymptotic, well, in the same months when this asymptotic expansion was done by Gopakumar and Vafa Faber and Kanderipande studied the application of the virtual localization formula to the case of essentially the the local neighborhood of the rigid p1 inside the Kavya 3. So lo and behold theorem fg of s3 comma t coincides with the generating function of Gram-Witton invariance by genus g for some Tori-Kavya 3 fold x where x is the total place of 2 copies of the tautological line bundle of p1 so left and side of Gopakumar and Vafa and for example if you look at what happens when you compute for example a planar limit of the color of the polynomial in the case of a single winding number so it's a function of Kappa and t is some hyper geometric function minus d df minus 1 df plus d minus df e to the t well there's some proportionality factor that I'm discarding here on e to the t it's related to the beta function that I put it there and this is a code skew and tau j function for x well, equivalent with respect to some torus action specified by the framing f so there is some framing here as well there is complete coincidence of two gadgets that well in the face of it have absolutely nothing to do with each other something leaves in three real dimensions that is a topological invariant and then you have a symplectic invariance of Kali-Bial trees and it's even, well it's hard to tell why you really have to be grumbling with in theory to reproduce the one of our expansion of transcendence theory and if you pick s3 what's the relation with x so y and many y's so let me address first though so how does this Kali-Bial 3 very simple Kali-Bial 3 arises from s3 this is not the simplest Kali-Bial 3 that you can construct out of three manifold that would be the first shot at a six dimensional symplectic manifold that you can construct from s3 is well you take a cotangent bundle which is this can be equivalently represented as dfine quadratic in c4 suppose s3 as radius r dr would pop in here as a complex deformation parameter then you can regenerate this s3 what you get is a quadratic cone so this is a cone over s2 times s3 and this admits a canonical torque resolution which is prevalent and is given by the small blow-up of this cone and this is a vector bundle on p1 which is to p1 is actually rigid and there's only thing that this can be this is o-1 that's o-1 there's some relation between the two let's say o-1 so physicists first gave an interpretation in terms of well a copy space interpretation now this iterating function of ground-wetting variance have an interpretation in type 2 compactification as computing some protected terms in n equals 1 n equals 2 and for example there's superpotential superpotentials can be computed either by considering topological string with either brains, so some open string theory leave it in here or by considering fluxes and turns out that the brain configuration in here can be related wisely, this was done by Vafa, lifting to my theory Atya Malthusen and Vafa so that you get an identity at the level of what this computes in the physical theory and so microscopically another thing that would be desirable is to well see some theory of teacher assignments somehow appear from the string at the integral and the result comes and there's a beautiful paper by Vafa which builds heavy on previous paper about wetting so we're going to show on that open ground-wetting variance on T star of S3 with a Lagrangian taken as the zero section of this bundle is equivalent to Simon's theory so this boils down to proving an equivalence between open ground-wetting and closed ground-wetting and in physical terms you can see this as a sort of new phase that opens up on the worksheet when you integrate it out it creates holes on your worksheet and that's precisely what you get so mathematically it's the sort of mathematical version of this phi2 in some ongoing work of YPLE which is pretty much along these lines and second mathematical or physical depending on your inclinations explanation comes from the fact that both sides are governed by the topological recursion in particular, Chen's Simon's theory can Chen's Simon's theory partition functions have a representation an integral representation due to first Razzanski and Lawrence and then more generality by Marino in terms of U and matrix values and this gives you the kind of structure that you expect from a topological recursion W10 and some higher WGH computed by the topological recursion itself this has been proved actually by Mark and Laurent Tan and on the string side you have a natural candidate for W10 that comes from Toric Neurosymmetry and the last rest that you have to prove is that this gravitational descendants obey the topological recursion and this was what was into that at the end of their trans-talk so you have the same initial datum for the recursion and the same recursion the two things agree you have a conifold transition or resolve conifold to have torus-section but on Chen's Simon's you ignore torus-section it's kind of strange that's the thing, you have the framing that's put in torus-section to of Chen's Simon's so the thing is so normally you have a torus-section so in a constant approach you need to pick up a torus-section that preserves the canonical bundle of your x plus constraints on where you put the Lagrangian this leaves you with one parameter ambiguity but that means that Chen's Simon's also should desecrate a parameter with torus-section well what the equivalent parameter matches with on the Chen's Simon's side is process loop framing parameter so that's it's kind of strange because it is not completely intuitive it is on the two hands so you've got framing on one side and the choice of torus-section and it too happened to match but I don't have a conceptual explanation of why they have to match so in canonical framing you have some you have an identification and you see that changing framing amounts to changing torus-section on the other hand on the Gramm-Witton side but I cannot offer you a conceptual understanding of that so this is very nice because you see that two theories as different as these two enjoy a relation and there's something that you can do like this so implications Witton-Rochetkin-Turayov Gramm-Witton so on one hand Witton-Rochetkin-Turayov events are naturally defined at finite level and especially at finite rank so this resolves Gramm-Witton generating functions to all general typically something that is very hard to attain and one of so by gluing there's an application to hydrogen's Gramm-Witton invariance of Tori-Kavyaus and for example this is the natural gadget that you would use to test stuff that works when you want to analytically continue in the string coupling constant in the genus constant a Gramm-Witton DT and on the other hand what Ruben was was mentioned before is that there is a reinterpretation of Gramm-Witton invariance in terms of BPS state counting and so Gramm-Witton invariance are in general initial rational numbers they're not heavy-media an immersive meaning but in this context of Kavyaus they can be reconduced they can be expressed in terms of inter-world BPS invariance as shown by Pandarek Pandan-Thomas so this is kind of unexpected when it comes to looking at the structure for example of the homely polynomial for a class of knots like Brownian knots the LMO-V conjecture which was proven by by Kevin New and Pan-Pang about five years ago this is extremely appealing but it's one example there are many things that need to be understood and these that you want to ask is well if you cannot explain at least try to generalize as much as possible this picture from S3 to the L0 and gauge group Q1 to well as general as possible free manifold M and arbitrary knot and arbitrary gauge group G G has to be sorry in your expansion you have seen a large N limit so G is classical so UN S1 SPN so these are the case that you can find so in particular this extension to our trigonal-symplatic group was considered by Senior and Vafa there is an analog of these integrality conjectures one one extension to so moving from beyond knot to for example torus knots well there have been proposals for an A-model and B-model so both of a special curve and of some A-model target about Yakonesi, Shen and Vafa and for the B-model both myself and Art and Maurinho it seems very hard to go beyond this case of torus knots there isn't an ongoing big project though which seems very promising by Agnagic, Vafa, Ekul and Gunn to generalize this to the case of general torus knots but the role of the topological recursion in that case is at best unclear and a sort of more drastic thing that we can do is well change the free manifold and yeah this is what I'm going to discuss in the last five minutes so I think that well the broadest is probably the broadest class of examples to which you can put this the quality of transciences we're growing with to work is to consider M the spherical cipher manifold so gamma is a fine software there are two obvious questions Art what is the dual curve counting period that you're looking at second is what is the well using the translation for spectral curves there's the dual spectral curve depending on gamma such that well free energies and the gravitational descendants are computed from topological recursion on this S the general strategy is take the kind of transition picture seriously this is this gave us a big clue what to look at in the S3 let's try and persevere and consider quotients of the story by gamma there are essentially two cases the one that I want to discuss more rapidly is the case in which gamma is a billion this list of landspaces so you got a cotangent space on a landspace LPQ in here you got a quotient of the singular conifold by some cyclic group action so you started with some toric variety and you had a C star cube inside the quotient by a cyclic group you still have a C star cube inside and you can so this is still toric everything is toric in here and you can resolve it historically and it's going to give you some candidate for what X gamma should be for landspaces so the story is completely parallel to the S3 case when these free manifolds are monopled bundles there's a slight departure that I'm not going to discuss for the general landspace but this whole story can be generalized and in particular what's left is the spectral curve since everything is toric you have toric mirror symmetry that you can appeal to the whole above mirror is what will give you a spectral curve so this was done by many people starting in the beginning of 2000 the case that was left out is the case when gamma is up to central extension finding subgroup of S2 so this is what I'm looking at presently with an MPI bond and Albrecht claim now in this case gamma is not abelian so what you're going to get in here is something that is not toric there's no there's no horibuffa and no spectral curve that you can construct this way but with the fact that the subgroup is inside S2 you see that the quotient group acts fiber wise on the result kind of so what you get is you can resolve this fiber wise by the canonical resolution of type ad singularities it is going to give you some people which is not toric but that is absolutely it's clearly identified the problem is is there any way starting from there to construct a spectral curve and one way that is inspired from the geometric engineering of Katz, Klein and Wafa is actually connecting some dots in the physical literature is that well the spectral curve for this gamma is the spectral curve of the well I'm going to call it it's probably not standard notation but the ad is relativistic actually the generalization to arbitrary lead groups of the periodic relativistic toric so for type ad this was considered by Reusenars in the mid-80s and then generalized to arbitrary gauge to arbitrary in today's groups by Soresen most recently by Fokker-Marshkov this passes several tests and in particular since in this case as well as for land space and the stream the Chernzymons observables enjoy a representation in terms of matrix integrals you can match the spectral curve that you get on one side with a proposal that comes from here we don't have a complete proof at the moment but for example for the trigger connection contribution we can test well we can explicitly see that the spectral curves of the Chernzymons matrix model is given by a suitable restriction of the actual variables of the spectral curve that you get from this class of complex integrals systems and if I can conclude in two minutes this one application is as to once conjecture to the Krapman resolution conjecture a higher genus this sort of philosophy was exploited in Taipei by myself with Kaolier and Ross to prove that the higher genus full descendant version of the Krapman resolution conjecture holds for Taipei singularities by which I mean that the full descendant grown with the potential for Taipei or C2 mod ZN resolution is given by the action of some quantis operator quantization of some element of the symplatic loop group acting on the full descendant Gromwitten partition function at all general of the orbital theory and it was crucially there to explicitly see that the R matrices in the Gaven-Tals approach for the calculation of the ZS agree on the semi-simple locus and this comes from some asymptotic or some asymptotic in one variable for the guide you extract from the spectral curve and for DNA the kind of Gaven-Tals mirror symmetry theorems fall short of giving an answer but this integrable systems viewpoint is probably the key to to generalize this to the case of general AD surface singularities I can stop here