 Thank you very much, and thank you also to the organizer for giving me this opportunity. So, yeah, in my tato you see three words, but in fact what I really like is the resurgence, the middle one. And what I would like to understand and also try to explain today is a kind of idea how resurgence and how much resurgence knows about modularity. And, yeah, so in particular what I was working on, also discussing with Maxime and more recently with Claudia Arella, is studying resurgence of divergent series, which arise from some Q-series, and then trying to understand if the resummation is in a sense related to quantum modularity. And in fact more generally what I would like, what I will propose in my talk today is an attempt to give a general picture where you can expect quantum modularity, so a general resurgence structure from which you maybe can always find quantum modularity. But actually in my main example of this talk we would rather go in the opposite direction. So in a sense the example I will consider, we already know that there is a quantum modular form. And so the question will be more like can this modularity property tell me something about the resurgence structure associated to a certain Q-series. But before going to the example, I would like to start like briefly calling some idea related to resurgence and Borella plus some mobility. So as we have already seen in these workshops several times, very often we deal with divergent series. And in fact what we would like to do is to extract some like analytic function out to something that behaves not really nicely. So in particular a series was typically coefficient growth factorally fast. And the first thing you can do if you encounter these divergent series is first to consider a Borell transformer which is a purely formal procedure that divides out your divergence. And this typically give you a germ of analytic function in this variable zeta. So here you are in what is called the Borell plane, so the zeta plane. You have a finite series of convergence and you may have these cross are the singularity. And so what you start to do is now trying to do the analytic continuation of this function. And the hope is to find like a region when you can actually take this ray that goes through infinity in some direction and where the function is actually analytically but not only that it also must have some nice decay condition. Because if that happen you are allowed to take this Laplace transform and as a result you will get an analytic function, analytic in some half plane related to the direction of theta. And what is the relation between the analytic function you get and the original divergent series is through what is called Gevrea Synthotic. And then I mean this will be an analytic function in this sector but when you start varying this angle actually you will understand how to also extend this analytic function. And every time you will cross one of these singularity, typically this will jump and you will see the stokes phenomena we have also discussed in this talk. But in fact so this is called the like paradigm of Borella plus some ability. But sometimes what I think is interesting to consider is also example in which you actually have an analytic object at the beginning. So you actually start with some analytic function and then you study the asymptotic. And it may happen that the asymptotic is a divergent series. This is for instance the example of the gamma function that Maxime presented or more general for exponentially integral this is what happened. And so the idea is okay now I got something divergent. I can try to apply again my paradigm of Borella plus some ability. But so I will get another analytic object somewhere and are these two things equal or somehow how do, how are they related in some way. And in fact what you know prior is just that they have the same asymptotic but this somehow is not enough. And we have a good result which is the vanilla theorem which actually tells us okay yes these are going to be the same if you have a good control on the asymptotic. But more generally what you should do is first kind of relax the definition of the Laplace transform you want to consider. And in fact in the example that we will see today this is not really enough. So in a sense if you start with, if you have at the beginning an object which is quite nice maybe not exactly analytic but in a sense more regular and you start computing asymptotic and doing this procedure you will not get back exactly the same object. You have to modify it in some way. And maybe I should also say that another example in which you can instead close the diagram are in a sense exponentially integral but also solution of linear ODE. But okay so this is kind of the true paradigm that we may consider and where does the resurgence come from? Precisely in a way you can relax this definition of the Laplace transform. So here is like the definition of resurgence function as David also explained us today. So I'm not going to read it all and explain what I really need is if you want the first line so a function now I'm in the Boral plane it's resurgent if it can be endlessly analytically continued. And in fact dealing with the fact that you want to end this analytic you can end this analytic function. In a sense it tells you that you can allow more general set of singularity so in a sense the set of singularity can also be dense but because as David was explaining yesterday in fact they live on different sheet of your human surface. And moreover what is important to me is that the Boral plane in a sense knows all the information meaning that if your goal is always to find some analytic object behind your divergent series once you apply the Boral transform if you get something which is resurgent it means that you can study the Boral plane with the tool developed by a call and this already contains all the information so it will tell you how your analytic function will jump and it will be defined on the original H bar plane and this is how these alien calculus and also media Laplace resummation or all this version in a sense of Laplace transform are more powerful and very well defined to work when you have a resurgent function. So my point is really if you have a resurgent function essentially really what you have it's a great knowledge of the Boral plane and all this machinery helps you somehow to avoid all the conditions that you need in order to take the usual Laplace transform you can actually deal with much more general object. If you hide the nozzle of Laplace transform, what is HET? HET is the Boral transform Ah yes sorry, before it was G, thanks. Okay, so let me give again an example of what is called a simple resurgent function again Maxime already gave this and it appears several times this week so if you consider a singular point of your initial germ G this has a simple resurgent structure if it has the following behaviour so in particular you may have a simple pole and then you have a log, time and other germs and maybe a plus a monomorphic function and this structure is quite interesting because in fact the idea is at the beginning you have your finite rays of convergence you find a singularity if at the singularity your germs behave like that now here you can start playing with this other object with this other germ and so you can start to study the singularity of this phi HET and if you find another point this will give access to a different region of the Boral plane and you can continue studying all these germs and in fact if your function is resurgent in a sense it's like that all these germs know each other so for instance if you only have a finite number of singularity as in the case of ordinary differential equation you can consider for example the IRI equation and you will have only two of these germs and if you once you have found the first one you expand, you find the other one and that's it and if you consider the second one and you expand again you will get back the original one so there is really a very nice structure but in fact what is important here is that all the ingredients you need to reconstruct your analytic object are already there and in particular are this constant CNS as well as this germ and why I say that? because if you now want to go back to the original H bar plane you are going to take some Laplace transform and in a sense these will give you just a constant but here you will see that this log will give you this constant as well as this germ so this is like the information that also Marcos Marino was considering speaking about the resurgent structure so since this is the important information that we need let me just recall the definition that Marcos was saying so a resurgent structure for me will be a collection of singularity, this set omega a collection of germs and the Stokes constant and for Marcos the only difference was that he puts together like the collection of germs with the singularity so in a sense it's just one function and I'd like to keep them separate because in a sense I am kind of implicitly assuming some normalization condition so in particular I always want this to be integer and now I'm going to introduce a sort of proposal for a different type of resurgent structure what I will call a modular resurgent structure so essentially I specify the data of the resurgent structure in the following way so I would like to have a tower of singularity here you have an example, it's horizontal and not vertical but the point is that all these points like are separated by some integer m and you may have infinitely many all the germs are one so in a sense it's a quite simple resurgent structure I mean even simple than the one of before because either you have only log singularity or actually only simple pole and finally I would like to have that my Stokes constant gives me an L function when I write them in this way so I do not specify which type of L function I mean there are many, usually it's maybe directly L function RTL function I leave it on purpose a bit vague because we will see an example we can have both and so far let's leave it in this way why this is a nice definition is well because in fact you can construct an example in which this happened to be the case so in which you have this modular resurgent structure and here you have an example so for instance imagine that you consider an L function of the type you like and also you consider a set of points like a tower or okay as I was writing before and then you construct your divergent power series so BGV1 means that this coefficient have to grow factorally fast and you assume that the coefficient actually look like that so up to a constant you have a factorial growth and then you put this sum okay now the claim is that if you consider the Borel transform of this divergent series it has a modular resurgent structure and what does it mean? it means that the Stokes constant will be exactly this SK and the position of the singularity will be exactly this ROK so how do you check this? in this case it's really simple you consider the Borel transform meaning you divide it by this factorial and this gives you an N and you change the variable and then the only things you have to think a little bit about is you want to exchange these two summations so actually this is also an infinite summation in this case you can actually do because thanks to the Geveret property you know that this is going to be something convergent so if you permute the true sum then you will get the geometric in this case actually you will get a log up to some correction so you can recognize here exactly the same simple resurgent structure I was discussing before so you don't have any other function here so that's it in a sense you only have this log singularity you see that the position of the singularity is exactly the ROK because it's where this is as a singularity and the Stokes constant are precisely written in front of it up to some normalization this is a normal function it's just because I would get a zeta divided by ROK and so I just put maybe this is just a constant I think I think it's just a constant yes yes okay so this is an example actually you can do also something slightly different so in a sense you can replace this factorial with maybe a gamma shifted by some positive rational number so if you again you do the same game now you assume that your coefficient look like that ah yes I would say one means that the coefficient cn grows factorally fast so maybe a to the n n factorial for the n large and yeah okay so even if you consider something slightly different you will get again a modular resurgent structure and you do exactly the same computation the only difference here is that you don't have any log you actually have some square root or like different type of singularity and so you can still see that the singular points are ROK and about this constant here to see that they are exactly this as k you have to like compute the monodromy so you just do a loop around the singularity and in fact they will be up to some maybe different slightly different constant again multiple of sk so even in this case we can say that the stokes constant are always exactly the coefficient of this function so we have these two examples but why we should care about this modular resurgent structure so in fact they appear in this context but even more what I would like to prove and this is a sort of general claim is that if you consider a divergent power series with rational coefficient which is jevre1 and with borol transform admit here I am adding simple modular resurgent structure and by simple I mean that I have only simple pole or log singularity then taking some generalized Laplace transform so meaning that I take this integral but on purpose I do not specify the path here then it defines some quantum modular form so in particular if you take f tau to be this generalized Laplace transform but now where h bar becomes 2 pi i tau this is a quantum modular form meaning that if you define it for tau rational you have this type of transform so f of tau plus 1 goes in something like that this m should be related to the distance I have in my tower of singularity and then for some SL2Z transformation you would expect that if you take the difference of this and this other function what is here this h gamma tau which is usually called the co-cycle is going to be defined over r smooth but real analytic except a finite set of points typically the pre-image of infinity or maybe even you have to remove maybe even 0 so the idea is here again it's not really precise and in fact I should say that in the example of yesterday in William talk what he was actually he has a theorem in which this path here it's the one that give you the median Laplacere summation so it actually in his example he have a tower of singularity somewhere there and the path he was considering was something like this but I think as we will see in the example later on it doesn't look always the case to have this simple path here so the other remark I want to make is that I think almost I mean at least in the example we consider it's not enough to take the usual Laplace transform where the usual Laplace transform now my tower of singularity usually is here so the usual Laplace transform would be maybe integrating over there this is typically not enough ok so are there any question about that what is the weight in your weight one and indeed I think that in the example instead of William of you and your collaborator it should be fractional weight and I should also be related to the fact that the coefficient has a different growth so you don't have simple singularity but I guess you will always have this fractional power if I remember correctly this bad at series actually has also the Borrel transform that you wrote down yesterday has maybe a square root singularity if I remember correctly yeah that's right so yeah I would indeed maybe one speculation one further speculation can be that in the case you have a square root type of singularity you will get fractional weight but in that case maybe the counter will be precisely this thing so you will always do the many other summation maybe for fractional weight I guess it could be reasonable that you always get this many other summation and I have another example in which you have the fractional singularity and indeed the many other summation is what works well ok so maybe now I will go on with the example and so my main example is like the one I was discussing with Maxime and in this case we have we know that there is a quantum and modular form so here I just try to recall some of the results we are knowing before due to these people and so first we start with the Q series so actually in this case we have two Q series sigma Q and sigma star Q where do they come from actually was Andrews that considered them related to some venatoric problem like counting partition but I will just look at the two expression you have on the right hand side so in particular these are well defined and convergent inside the unique disk but most importantly they also make sense when you consider Q at root of unity and in particular they are actually related by this expression you have this equality and then what Cohen did was actually to consider to define some coefficient Tk as the following coefficient so the coefficient of these Q series and of the other one with the sigma star and what he proved is actually if you consider now the sum of all these coefficient divided by k to the s of an artinal function which is defined in this way okay so we start in a sense like we start seeing some of the ingredients we may need later for our to build our resurgent function but in fact most interesting that here in his paper about quantum modular form he defined a function f defined for Q equal to Tk in particular f can be defined equally as this series with the sigma or with the sigma star because of the previous equality and what he proved and here I add Maxime because he realized there was some mistake and so there was an absolute value missing in the original paper so it proved that this function satisfies this true relation in particular the second one means that if you take the difference between this and this function this e to the 2 pi i24 f tau this co-cycle it's real analytic it's smooth and real analytic outside these two points and I mean this is something that one can easily check from the definition of this Q series this part here in Zagir paper was proved using mass form but you also using the property of the L function over there in particular the functional equation that this satisfies and what he does is like to write these h tau in terms as an integral of some mass form so it's purely analytic it's purely analytic proof and now okay that's nice but what we do now it's time to introduce the divergent series so what we consider is now a formal power series which look like the f tau of Zagir but now where tau is h bar yes tau is h bar divided by 2 pi i or maybe h bar is 2 pi i tau actually even more general so now h bar is just a complex parameter but this is just to make the parallel use definition and if you expand these like we just add an h bar over there but okay if you expand this numerically you see that the coefficient are rational and moreover that they grow factorally fast so actually from now on also the conjecture that you will see it means that everything was check numerically with a very good precision but unfortunately we don't have a closed formula for this coefficient so the first thing that we check numerically is this type of relation so the coefficient here grows factorally fast and you will see this tk the same as before here is coefficient and divided by this rho k so now these as you may expect if you believe the conjecture will be the position of the singularity and these the soaks constant and you can immediately see that they are like in a tower like that it's a bit fast for me you define the formula series by extracting the Taylor series from that series of functions but as a series of functions does it converge somewhere? nowhere no no no it's we just put this in the computer and we start extracting the coefficient and you see that they grow like this I mean if you want here you can first put like an asymptotic like for large n they look like this when I was looking above this cn I mean it's just rewriting this formula so if you just it's adequately convergent it's a formally convergent series each of these formula series is a function and I'm asking whether there is anything analytic I mean do they make sense as a series of functions somewhere? no I mean you need disk but here on the boundary you need disk inside you need disk it's okay so you are extracting the asymptotic extension somehow okay okay so if we now assume that the coefficient are exactly of this form we can compute to Laplace transform in particular in the vertical direction and in the vertical and the lower direction you will and this will define for you true analytic function one in the upper half plane and the other one in the lower half plane and in fact if you compute the discontinuity so you can understand how they jump across this ray and what you will see it's just a computing residue so in particular like on the right you will get exactly the singularity that you have on the right and if you do the other way around you will see the negative one and these are actually also the trans series type of expression that we have seen many times this week so in particular you have all the exponential with the position of the singularity divided by h bar and these are the coefficient sorry rho k was k pi this pi square k over 12 yeah and moreover if you look at this series actually they more or less look like what the series that you find the coefficient tk and in fact you can rewrite them but in terms of q tilde series where q tilde is e to the minus 2 pi i tau sorry divided by tau so if you just specify h bar to be so if you use this change of coordinate sorry this transformation h bar equal to 2 pi i tau you get the back this q tilde series and the q tilde series also appeared in guk of toky yesterday and you will see them later on so in a sense you have this type of transformation when the stocks constant are in a sense and actually the discontinuity can be actually packed in a q tilde series which is again kind of related to the original series sigma and now the second conjure so we have this Laplace transform the question now is is it true that my f tau is equal to this Laplace transform when h bar is equal to 2 pi i tau and let's first look when h bar is actually 2 pi i over n so again this is check numerically and what we have is that 1 over n f of 1 over n depending of whether you take positive or negative look like the Laplace transform in one direction or in the other but there is a correction and what is a 0 is 0 is simply e to the exponential so e to the 0 of x is e to pi i x so it's not exactly the same there is something more and Now you may wonder well what if instead of 1 over n maybe I just put tau because at the end of the day I want my f of tau for tau rational. And if you do that actually again numerically you see that they do not match. So if you replace everywhere 1 over n by tau and you start testing this equality you will see that are something completely different. So it seems only this written in this way it's only valid for 1 over n and you will see in a while also why it's true. In a sense this is related to the quantum modularity of f tau. So okay it's kind of strange and so let's try to look better of what quantum modularity is selling as though. So in fact we have the following, so we can actually simply compute this, check this expression over there. So in particular remember that the cycle was defined in this way. You take the difference on the two. And now if you do that again you can check numerically by now I will tell you also how do you obtain this. You have that disco cycle for tau like bigger than minus one half and not zero has this expression and what does it mean now maybe I should. What is E1? E1 is just a way to rewrite the Laplace transform. So it's just integral from zero to plus infinity exponential of minus 2 pi t dt t plus ax. So just a way to rewrite the Laplace transform but this is somehow related to the exponential integral function. Okay so you have this expression but what is interesting in our sense how do you prove it if you believe the previous conjecture and the idea is the following. You first look at this expression for h but with tau equal to one over n. So you rewrite exactly the same thing putting a set of tau one over n and then you use the previous expression for f tau written in terms of one over n. You will get exactly what is written there with tau replaced by one over n but in fact the surprise at least to me is that if you now use tau whatever rational number now this is correctly numerically looks numerically correct. So in a sense I use a conjecture that only is true for one over n in a sense but if I go back to tau that's worked nicely. And okay in fact even more you can actually playing a bit with this formula again numerically check and obtain the following expression. So now this look in a sense much nicer and in particular what I learned from this formula is that my tau f tau is kind of Laplace transform here plus and minus is just because tau can be positive or negative so I put either plus or minus depending on the sign of tau but you have this extra thing here which in a sense come from there. And what is nicer is also that everything here comes from the previous conjecture the one with f of one over n but in fact you can also assume this to be your main conjecture and if you replace tau with one over n you will get back the previous one. So in a sense this term here will correspond to what before was the terms with a knot the other correction so maybe I should just move okay here before for one over n you had this is zero so this exponential of 2 pi i times n factor and the reason why it was not working in this way is because in fact if you put back any rational the formula became different and what I mean and to fit it in my previous kind of statement attempt I do not I mean I would like to write it as a Laplace transform maybe generalize with a different counter but the point is that I don't see why the right hand side can be something like that like with a media resummation type of thing so maybe it is maybe I have to do something slightly different I mean I don't know yet so far this formula it's telling me that I have this this extra correction but it really comes from the quantum modularity because it's like the different of the I mean yeah it comes from the definition of the cycle in a sense okay so if there are no question about this example I will go on with the other two yes f was defined using my series so the sigma q so f of tau was q or maybe just write e to the 2 pi i tau divided by 24 times the series sigma q with q is e to the 2 pi i tau and yeah sigma q was a series involved the pocummer q pocummer yeah you didn't roll but one can this the version series is rational coefficients that you make in the bargain actually appears in one of this formula it's not a very simple one-dimensional formula which I introduced yes yes one of the simple yes dialogue what I'm developing with some power of law square of law yeah indeed I mean the reason why we got to this example it was because we were simplifying one of the ones the formula Maxine gave in his last talk okay so let's go with the other example which okay yes it's the concept it's a gear powers in Q series okay so again here a bit a quick recap of again this is also an example in which we know there is a quantum modular for so the idea is you Maxime actually consider first this Q series now this time this is not convergent either in the unique disk or either inside or outside the unique disk and but it makes it is well-defined at when Q it's ideal to unity because essentially the series terminate and what the gear notice actually proved is what he called the strange identity in his paper about vasili of invariant so the statement here is that these Q series is actually equal to something which is on the right hand side but the right hand side now is a convergent series in the unit disk so what does this thing mean is that if you take the radial limit on the right hand side you will get something convergent to these power series and okay here the coefficient on the right hand side are given by this character here so this guy and is written over there so this is our Q series and in fact what the gear also proved is that if you define again F from Q to C in the following way so you just add a Q to the 1 over 24 and you always use this definition for the Q then these F tau satisfy again the following relation so this is the simplest one if you just put if you just translate tau you said tau plus one in this formula this is more interesting now here the co-cycle is defined by taking F of tau and then you have here something that you see it's fractional I mean it has this fractional power F of minus 1 over tau and it's as it should be real analytic except except the origin where this is a problem as a problem okay so as you may guess now this will be an example this is an example when you have a non-integer weight yes but H actually it's defined over C over R sorry you'll see okay and so now the question is where again this is fine but let's look for something divergent our divergent parents power series which was actually considered first by costing anger of Alidis so as we did before these series you just replace 2 pi I tau by a generic complex H bar parameter again these as rational coefficient but this in this case we are happier because the coefficient as a closed formula and this was already proved in the paper by Zagie and so what Karouf Ali this and constant did was okay let's compute the Boral transform of these function here the of these divergent series and the first thing then they prove is that the Boral transform as this expression so in particular you see it has fractional power singularity and you also see the position of the singularity which will be at this K square pi square over 6 and if you compute the discontinuity what you get is the following expression so okay here there is an H bar with some fractional power but in fact these will be the stokes constant okay and okay is just what was before so this is another example of my of what I was defining modular resurgence structure because the singularity are all real and actually real positive so you have your tower of singularity and these K times K K I didn't said before but this is related to the dead in Kinecta function. What is XI? What is XI? Kinecta. Kinecta sorry yes it's it's this one yes okay and in particular if I if you so okay let me just in fact you can actually rewrite this coefficient and you can see the gamma appearing so this gamma of n plus 3 half as well as here and again from the two formula as I was just saying you see where are the singularity and what are the stokes constant so it's again an example as I was saying and in this case you know where there is a quantum modular form but in fact what the I mean the main interesting us I mean what is really interesting in the story is that now if you take what is called the Mediar summation you will get it back exactly your F tau and this is non-umeric nothing everything is precise and here I just rewrite so if you want these some on the right on the left hand side is this Mediar summation which is exactly of the type William was saying yes it was considering yesterday so you have these two paths and another way to think of it is that you as William was saying you take the path which is downstairs you move it above and doing that you will collect all the residue from the singularity so in this example you see that in fact is the media resummation that works as also yesterday was claiming William was claim but yeah I have this a special that is somehow it's related to the fact that you have this fractional modular form I think that for integer may be different the the function that you have to consider I don't know but okay yeah so this is a kind of this was the other example and again everything is exact so you don't have really to do any numerics or nothing and maybe the other comment I should say that in order to prove these they actually also use the strange identity of the gear so once you have the strange identity and more or less what was in the previous slide you can actually realize that this is equal to f tau so yeah in that case actually you also use the strange identity which in general maybe we don't have okay so now if there are no question about the example I will go on to the last one which is a one which is the one in progress also with Claudia so actually what I'm going to present in this last example is an example in which we don't have now a quantum modular form on the nose so we don't know whether there will be or not but we will see that we have this modular resurgence structure and so what we are trying to do with Claudia is to construct the quantum modular form okay so where does this and actually this example come from from something that has nothing to do maybe with the series we see this week with knots invariance and so on so Claudia in in her paper consider this trace of quantum mechanical operator and here I'm just looking on the example of local p2 so exactly the example that Pierre Rique were discussing before so what she computed is this expression where you have a bunch of q series and also q tilde series okay and in particular so already from there you can consider these are well-defined both as Q and Q tilde series but in fact in order to get these divergent series she considered two limits the limit at one Q tilde going to one or Q Q or Q tilde going to one and she get two divergent series in the parameter h bar and here I have to say that I keep using my convention H bar equal to 2 pi I tau but in fact in our paper the convention is likely different and this because H bar it's a physical parameter so it's reasonable to have it like parameterizing a different way with respect to tau so I just change it in this way because then we will see more clearly like the parallel with my convention as before my definition as before but okay if you look at the paper is slightly different and so these also mean that somehow my H bar is not anymore the physical parameter and these two limit that we have here again actually have a physical meaning and they corresponds to two different points on the modular space of the local club you know okay but so now we just look at the Q at the divergent series which has which are inside of this exponential sorry so we have three equalities with trace of rule the first is the function is defined this way and yes and the last two the limit is asymptotic expansion yes that's true yeah so they're not equalities I know sorry so asymptotic expansion as H bar times 2 0 or infinity exactly yes okay and I mean this case is nice because again we have closed formula for the coefficient coming from the the book cover so yeah everything is exact okay so now here I just put all I mean many of the result that Claudia proved and only the one that in a sense I need so we consider this divergent series this is a one at zero so this is why I have this G naught this is the Bernoulli polynomial number and this is the Bernoulli polynomial computed at yes they are a bit too close yeah okay so this singularity in this case lies on these four pi square k and the stocks constant satisfy I mean gives you an L function which in this case is this product of the Riemann Zeta and the Dirichlet L series associated to this character shifted by one okay and in particular she also proved that locally at close to each singularity the Borel transformer has the following expression so in particular is a simple resurgence structure you see that you have no other correction and the singularity are exactly at this Rokei and the stocks constant as I said are the SK as in particular you say that Borel transformer is an analytic function and Rokei are all for singularity okay and moreover so okay in particular this is a resurgent resurgent but modular resurgent structure and moreover what she observed is that you can have you can rewrite the coefficient in the following way so exactly the way I was writing at the beginning this is what she called the exact larger the relation and this comes from the fact that you don't have other series attached to the log so since you have only this one this became an exact larger the relation otherwise you may have got more coefficient and computing the discontinuity now this Rokei you have them only on one on the real axis on the positive real axis so if you compute the discontinuity the discontinuity you will get the same type of trans series expansion where you have the constant and the Rokei divided by each bar but you see again that this is kind of proportional to the Q tilde a Q tilde series and what is this Q tilde series is actually part of what we have at the very beginning so they initially the trace as true QC oops here yes as true Q series you get exactly this one over there and there is the log because the series I'm considering has a log I took the log of the original one so you have this nice expression and for the other series the one at infinity it's kind of the same so again you have this nice exact expression for the coefficient and the Borel singularity are this point over there and the Stokes constant again satisfy kind of the same relation in I mean it more or less the same function but shifted you have just L of s and the Riemann zeta at s plus 1 and locally again you see simple singularity no other term and yeah Stokes constant and S k and Rokei so in particular you have again an example of these modular resurgent structure and thanks to the fact that you don't have any other series you have this exact larger the relation which now okay you have this gamma of 2n minus 1 but still like integer and if you compute the discontinuity you get this log so the this log where you have the other two Q series because now we are at infinity and so we see the Q series ah sorry here there is a type I think shouldn't be divided by h bar but multiplied by h bar because it's the one at infinity that's okay and so now I mean it's still kind of in progress to figure it out what is the quantum modular form in this case so if there is a quantum modular form and the expectation is that it should be related with this difference of logarithm so that this this difference of logarithm should satisfy the same type of transformation I was writing before but again it's not in this case it's not clear whether the medium Borelli summation will produce the the L function you want so yeah kind of open so far okay so yeah I think I'm gonna finish maybe a little bit earlier but it's fine so some are rising okay not yet the point I want to make is I want to make is that we have introduced a bit naively this modular resurgent structure but I think that it seems promising at least in some example and even thinking about William talk more generally to produce some quantum modular form so in thinking about the example of conceiving to the gear and William it seems that these general generalized Laplace transform should actually be the media Laplace resumption because this is the theorem that we have in some example but I think for what I call simple modular resurgent function so the one when you have either simple poor or log it may be different so it may not work exactly as in the previous class and yeah okay of course I mean this is kind of maybe naive approach but I it seems that in a sense all these quantum modular property can be already encoded at the level of the resurgent structure of the divergence series so we have seen examples so far where maybe one expect quantum modularity also for other reason like if you have this not invariant there are very good conjecturing which okay that sell you that there must be some quantum modularity and this is what William your David and collaborator were approved but from my point of view it seems that also resurgence amount knows of that so even like it's just something which you which is encoded at the level of the series and the point is how to it's understanding which type of Laplace transform you have to consider to extract the information so to really get the quantum modularity and that's what I want to say thank you very much other questions maybe your last example of this crisis are they related to the series which appeared in the previous talk to Perica a good question I don't know I don't know if it's exactly the same for he has these expression for when you want to apply WKB but I don't think it's they find it in the same way maybe you you can comment cloudy as well but yeah this is this trace comes from inverting the operator that comes from quantizing the miracle which he was showing supposed to be killed by the operator defined by the one yes yes I think the series are like slightly different different facts in one like the trace of these operators was the other one was about like the one I was considering was that the WKB solution of so yeah I think probably there should be some clear to me is that the order for point it's either each bar zero or infinity no I mean in the terms of the much less base of P2 is that generic so when you take each bar going to infinity that corresponds to the control the expansion and take each bar going to zero inside you're looking at a large radius but the modules of P2 is kept arbitrary this I mean these are points in modular space I understand but you're talking about each bar I'm talking about the complex modules so the mirror of the two isn't the queue isn't related to the queue I guess what before was that the tower some in the T in Terek talk or something like that okay so then okay maybe maybe we can do it yes may question you have some L function associated to this quantum model reform do you have some conductor if this comes from some out about geometry or some some motive no I don't think so in the example considered by maybe we I'm also the one of conservators that year you see that it is a directly a function but like in the example of Cal that you have this product the other one you have an art in L function so I don't know whether all this thing can yeah I don't have like a general picture for which algebraic geometry should correspond or motive yeah okay no more questions let's thank the speaker again