Resurgence and Modularity
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Gukov, Sergei
Why do modular forms tend to have integer coefficients in their q-expansion? Are they counting something? And, can we "categorify" a modular form by representing it as a graded Euler characteristic of some homology theory? There are several ways to answer these questions, e.g. the one based on classical theory of modular forms relates them to counting points on elliptic curves and motives of higher-dimensional varieties. In this talk, we will ask these questions for mock modular forms, introduced by Ramanujan. They also exhibit q-expansion with integer coefficients and, surprisingly, the answer to these question is based on resurgent analysis of a different power series, which at first shows no signs of integrality!
 Well, first I want to thank the organizers for organizing this meeting and in fact running it for a number of years as well as for inviting me here. I have to say that I am myself really new to resurgence at least compared to all the other speakers. In fact, 95% of what I'm going to tell you I learned from either Markus or Maxim and I want to thank them for sharing their wisdom, so that was very helpful. The only little piece which is new will consist of two concrete theorems that I mentioned toward the end of the talk and these two theorems come from two papers, one of which is with Markus and he's very bright student Pavel Putrov and another is from paper of Pavel and Kumrun Bafa. So in this talk I'll try to make connections between two completely different subjects, one which already appeared earlier this morning and the other one which has to do with modular forms. So I'll try to show and illustrate how we can go back and forth between different subjects. First of all by asking questions of one subject in the context of another, how the links sometimes can provide the answers to questions, sometimes quite surprising answers which are not expected if you're narrowly focused just on one particular topic. It's a good illustration that bridging two fields may actually help to understand structure of things that we're trying to study. And I'll also raise a lot of new questions by going back and forth and by thinking about this in a broader context will lead me to lots of questions that I'll now use as opportunity to ask during this meeting. In fact if any of you happen to know the answers please get back to me and I'll be happy to learn more. So like I said I'm new to this and I'll enjoy learning this even further. So I'll start with theory of modular forms and this will be somewhat pedagogical. I'll soon get to the place where I want to be but I'll try to frame it in a more canonical context so that if you're interested just in resurgence and have less familiarity with modular forms this won't be too harsh on you. So modular objects or modular forms specifically are defined as objects that transform nicely in this case under SL2Z modular group. So F is our function of variable tau and we require that as if we perform fractional linear transformation on tau F transforms in a nice way. Here a nice way means that it gets multiplied by C tau plus D to the power k. K is usually called the weight of the modular form and this factor is an automorphic factor that again appears in theory of classical modular forms. Later in the talk I'll show you somewhat more abstract or different versions of modular transformations where you can modify this rule in various ways. For example you can add additional terms and so on so depending on how you set up your story you'll get slightly different variants of modular objects called forms and used with other adjectives but this is the standard one. So let's continue from here. So our starting point is a function of variable tau and we want to characterize it. I also should say that here I'm focusing on standard Mobius group which is SL2Z. You can also look at various finite index subgroups or other modular groups. So at least for SL2Z the story is generated by two generators conventionally called S and T represented in matrix form as shown on the slide and one of them basically inverts tau is asking how function of tau is related to function of minus 1 over tau. That's interesting. That's in physics would be called strong recoupling duality but T does something a little simpler. It basically says that function of tau plus 1 is the same as function of tau and there is no contribution from this automorphi factor in front. So this second fact that f of tau plus 1 is f of tau immediately tells us that it's a good idea to first of all, expand and shape tau, introduce Q which is exponential of 2 pi i times tau and then this second rule will be implemented automatically. Also shown here is the fundamental domain and various translates our fundamental domain with respect to SL2Z elements. So for example shift by plus 1 takes fundamental domain here by minus 1 here and S does the inversion so we have this form of the cusp instead of going to infinity and so on. So this is how standard theory of modular forms usually begins and just like I mentioned a moment ago the T generator that tau is supposed to be same as tau plus 1 tells us that we should we better work with variable Q which is e to the pi i tau and in the rest of this talk it is this variable Q which will be playing the central role or in other words I'll be asking all kinds of questions how Q is related to various other types of variables that will appear and we'll be thinking about function of tau actually as a Q series as a power series expansion on Q with some coefficients a n which are priori are completely unconstrained they can be anything as long as this object satisfies this modularity requirement that under S transform or more general SL2Z transforms it behaves in a way we wanted to behave. So that's where it starts so again if you open a textbook and start learning this subject from scratch you may ask what are the simple examples of modular forms let's try to construct them and standard way to construct something which behaves nicely under the group is roughly speaking to sum over images with respect to the group or perhaps sum over lattice points. So then you can pick different entity which you're trying to sum over lattice points in this case say lattice is of the form of complex numbers which are of the form m plus m tau where tau is complex variable m and n are integers and you can try to sum image of the rational function or it's a translate set that there's different points which basically gives you this Eisenstein series so for each given k again remember k is the weight this defines Eisenstein series and you can nicely write it in terms of Q, Q expansion where sigmas are divisor functions these are sums of divisors to suitable powers for example sigma 2k minus 1 of n takes divisors of n positive divisors of n raises them to the power 2k minus 1 and adds them together. So this is our first modular form or rather simple class of modular forms which you can massage a little bit further you can do Lambert series resumption so on the right hand side we saw that in the Q expansion coefficients are given by divisor function this sigma of n coefficient of Q of n is sigma of n so standard Lambert series resummation says that that also can be written in a much nicer form by introducing denominators which are also functions of Q in fact in this you kind of get geometric series progressions which in the end resummon to this divisor functions. So the first non-trivial example is Eisenstein series E4 first of all we can quickly show that for odd values of the subscript you get zero because you're summing something and there are contributions with pluses and minuses so they cancel so first of all the subscript has to be even and then E2 is a very weird guy you may want from the start even before you start any analysis look at values of the subscript of this index greater than 2 simply for convergence reasons because you're summing over two dimensional letters and of some of a rational function whose denominators controlled by this index K so from the very beginning you may want to look at E4 as your first starting point and again just as we discussed in the previous slide this is how it looks like you can use this Lambert series resummation to write it in very nice form where in the numerator you have n cubed Q to the n and then you divide by this one minus Qn factors so after spending some time working with Don Zagir I learned my lesson that if you ever have to write a Q series or any function it's always a good idea to write first few terms so from now on in my life every time I write any function or any series I always present first couple of terms. You can't have negative powers of Q? No, no, no, never, at least in this subject so sometimes you may want to require the leading term the constant term to be zero in that case you get cusp modular forms and again you may impose further additional conditions which basically tell you how it behaves at the cusp that's very interesting that constrains the structure or modifies the structure a little bit but you definitely prefer not to go too far in a negative region at least in the traditional theory of modular forms. So this first non-trivial example of Heisenstein series E4 is quite nice and interesting again here is the first Q term so it's Q series expansion and like I mentioned earlier you can obtain modular objects by summing over lettuce and the Heisenstein series you can obtain by summing over lettuce in several different ways. One is by summing rational function with power 2k in the denominator that's how we introduced it. Another one is actually as a theta function where you sum exponential terms over lettuce point and here in this case if you sum over root lettuce of E8 you get a theta function which is in fact the same Heisenstein series. So this can be obtained in a number of different ways and has many interesting properties so this is very concretely if you haven't seen modular forms in action before that's how it looks like. Here is the next one E6 as I say only even values of the index or argument give you something non-trivial so lembert reclamation gave us that formula again following Don's advice I write for you first Q coefficients explicitly and here below in the plot I show how the real part of this function behaves it's very nice inside the unit disk and it has interesting behavior as we approach the boundary of the unit disk and later in the talk this phenomenon of approaching the boundary of the unit disk will play an important role so not exactly the way it's shown in the picture but in a closely related way. So that's our first two examples. We could try to go further and look at E8 and so on but one interesting fact about modular forms at least traditional modular forms and presenting for you is that the condition that we imposed from the beginning that object F under modular transformation is some factor times F of tau is multiplicative and as a result if you have two objects which transform very nicely with respect to modular group you can multiply them so they form a ring and in this case it turns out that the two guys that already showed for you E4 and E6 in fact generate the ring so the ring of modular forms is freely generated by these two fellows which means if I now start computing E8 it will be expressed as E4 squared E10 will be basically E4 times E6 and so on I have to get to weight 12 until I start seeing something more than one dimensional space of such objects. By the way did you notice something else about examples I just showed you? Integers. Exactly. So their coefficients are integer and this is kind of interesting because the original way we define modular form did not require this integrality. All we had is a requirement that it was a power series in Q that already used the T generator and then this power series in Q should have had nice property with respect to S generator which inverts tau but that did not a priori had to give anything with integer coefficients. So it's rather peculiar that the answer turns out and in fact in many cases in the theory of all kinds of modular forms the coefficients come out to be integers so natural question to ask is what is the nature of this integer coefficients? Are they counting something? Actually in mathematics every time you see an integer number it's counting something so the natural question is what is it counting in our case? A more modern version of the question is if you see a number integer number you can ask does it represent dimension of some vector space or maybe some cohomology theory? And in fact already from our discussion we see some hints for it that our integers are not necessarily positive or negative even the two examples I showed you indicate that there is no positivity condition so they can be just dimensions of something or counting something our numbers even if they have such interpretation are likely to be of the form of an index where we sum various dimensions with plus minus coefficients. So therefore what could possibly be is and that's a big question mark whether our Q series expansion can be written as a graded earlier characteristic of some vector spaces which are doubly graded and determines the sign and n determines the power of Q. So as I mentioned earlier in this talk I'll try to ask lots of questions to which I would love to know answers and maybe on this board I'll summarize my questions. These are real questions for you and I don't want to go astray and use much time of the talk for this but especially after please tell me if you have really good answers or good ideas and my first question is even in the context of modular form is there a cohomology theory which is doubly graded which provides which makes this a quality true? So the fancy name for this finding such spaces is called categorification. Anywhere in life where you see a Q series and if it has integer coefficients it's a good habit to ask this kind of question and again modular forms provide lots of examples of Q expansion with integer coefficients so therefore I want to ask is there a cohomology theory which really makes it true? In some cases it could be realized as not quite with an index but elliptic genus so it is an index but again I don't really know. Again if you happen to know the answer please tell me I would love to hear I don't know I honestly I don't know. In mathematics the closest I could find is et alchoromology but that's not exactly answering this question for example I cannot just say that H's there are et alchoromology groups unfortunately not so if again you know the real version of this answer please let me know I would be very curious and again I will try to ask analogous questions as we go. So with this frame of mind what are the integer coefficients counting can we realize them as graded dimensions of cohomology groups of actor spaces? Let's continue and in one class of examples of modular forms I will give you affirmative answer where the answer will be crystal clear and in fact it's not just exists it's actually useful for people in completely different area so in that case there will be answer and that makes me hopeful that more generally that can be true. So continuing with the crash course on modular forms we should go a little bit further and like I mentioned to you earlier the next interesting thing appears if we start going increasing weight of modular form in weight 12. So first we introduce dedicated eta function which is given by that earlier type product of 1 minus q to the n and this function by itself is already a pretty cool function. First of all it has q-series expansion as a sum and it has this theta like expansion with coefficients which I denote chi n these are characters of something of conductor 12 that is not terribly important but the same coefficients will appear later in one of the following slides but already this function itself is counting something cool if you take that infinite product forget about the pre-factor q to the 1 over 24 and just invert it then you see that you have 1 divided by bunch of geometric series progressions or factors which can be expanded in geometric series and that expansion will have positive coefficients and those positive coefficients in q expansion if you invert this infinite product is actually counting two-dimensional Young-Tablos or 2D partitions which plays very important role in many subjects from random matrix theory to modern gauge theory. Now in a course of modular forms dedicated eta function is interesting but what's even more interesting is its power 24 which gives you function delta called the modular discriminant and this is modular form of weight 12 this is not so hard to show by taking logarithmic derivative of delta of tau and like I told you before everything should be generated by our ring of modular forms is generated by e4 and e6 so delta should be expressible in terms of e4 and e6 in fact they're exactly two non-trivial well the space of weight 12 modular forms is two-dimensional so therefore everything should be combination of e4 cubed and e6 squared and delta is such a combination given by this expression here so you can multiply it out to get the first two terms as Don Seguir taught me to do for every function so to understand the meaning of this integer coefficients there is actually an answer in theory of classical form modular forms of course but it's not in the form I wanted it's not in the form of this categorification where you take coefficient and write it as graded holi characteristic it's much more contrived and not satisfying for our purposes but there is an answer and as an honest guy I'll tell you what the answer is. So Don Seguir goes via Borrell transform and if I were in Mil Borrell I'll definitely write reference complaint it could go something like this hey there is in a closely related context a very similar transform called Borrell transform that you saw earlier this morning which basically does this so given a function f of x you can produce it's in this case a number theory usually called Mil in transform for whatever reason transform which is a function of variable s and I want to point out that it is indeed closely related to Borrell transform of earlier this talk and much of what's what you're going to see during this day if you write x in an exponentiated form if x is exponential of some other variable then basically what this does is precisely the Borrell transform which you see in resurgence a lot of time so that's already a very interesting nice connection to the Borrell transform and integral also goes over ray in this case goes from zero to infinity and so on. So we apply this version of the transform Mil in transform to our modular form to obtain a function which again in the context of resurgence would be some version of Borrell transform that encodes some information about integrality or gives partial answer to this integrality that I wanted and what we get is a function and a number theory just like in resurgence it's a good habit to ask about analytic properties of this function it's a natural to ask a question like where does it extend in a complex plane where the singularities are what are the poles what are the zeros residues and so on. So this transform in this case specifically gives L function and there are several versions of L function one I denote by lambda and the other by L related by this gamma factor. Lambda is more natural if you want to ask analytic questions about extending this to complex plane and then in fact if you apply this Mil in transform to our class of modular forms or closely related objects that we introduced before you're going to find that it's an entire function and has two poles one at s equals zero the other is at s equals k where k is the weight it usually has some symmetry where you exchange the two and the coefficient of this the residue of this poles the coefficient vanishes if the leading term in the q-series expansion vanishes for example for cusp forms this would really be entire function. So for analytic number theory this lambda is more natural for algebraic number theory where we ask about integrality this L object defined by a prefactor gamma is more natural L is the Dirichlet L function of a modular form a modular object F and you can define it more directly simply by taking coefficients of the q expansion of our original modular form which was sum over n a and q to the power n and dividing them by n to the s so formally this defines some function of s and then again you ask same same type of question what kind of function is it what is it analytic structure that I briefly mentioned in the previous slide and so on. So in number theory there is an answer to to to integrality of these coefficients and namely they're counting something they're counting points on varieties over finite fields so first of all you take that L function you write it as product over all primes up to some finite set which may be related to a discriminant of your variety or maybe empty set in fact and you get something like this you have universal factor for for each prime and again weight k appears over there and when k is equal to 2 in a special case of way 2 what happens is that the number of points sorry this coefficients that appear there this AP's these are basically the same coefficients of our original series except that we're taking them at prime value of the index n they're counting number of points on elliptic curves over fields finite fields fp with p elements so AP is often called p defect because naturally you would expect that roughly speaking number of points would be same as number of points in this field plus one which is p plus one and a describes how much your answer is really different so the integrality of coefficients a at least for prime values of p and then multiplicatively for others are related to numbers of integer points over elliptic curves so this is a classical result which originally was a shimura to neyam a conjecture and then rediscovered by entree bay and finally made into mathematical theorem by andrew wiles and many of his collaborators and followers so this is pretty beautiful subject and it's interesting that the people involved had very interesting character alive so each of this guys is a quite interesting character unfortunately ten yama died very young so he committed suicide I didn't know this but preparing for the talk I checked the history of this on wikipedia and discovered that actually his fiance also committed suicide shortly after and that's actually very touching story of true love so that's that's quite interesting Gora Shamura is also interesting character and at keltec we had visiting committee earlier this week before I came here and digress told me about interesting discovery that Gora Shamura made in princeton so he spent much of his life and how big part of his life in princeton and princeton math building is very boring it's like a tower with many floors about 10 or 15 of them and it's it's uncharacteristic building all floors identical it's it's it's a very boring looking structure so princeton students made a very nice prank one day so they basically switched buttons one two three four and so on in the elevator so if you press button seven you would go to fifth floor for example so what else they did they also switched the office labels accordingly so you come out of the elevator and you go to all holes exactly identical so you go to your office and the office in the same place it has your label but then your key doesn't work so this morning security on campus got lots of calls from people that their keys do not work and Gora Shamura was the one to make a discovery that actually the levels were different simply because he decided to leave his office and take the stairs down and then he suddenly realizes that he has to go many flights many more than he would normally do so he made many interesting discoveries if you so here the while's theorem formally known as shimurritin yamakonjecture is about weight two modular forms which makes a correspondence to elliptic curves over finite fields but in in higher weight and in other variants of modular forms this is part of modularity program where you are not necessarily counting integer points of the full variety itself but of its certain piece called motif and that's an ongoing interesting research program so continuing with our crash course on modular forms in weight 12 as I told you before there are only two non-trivial generators so if you already have our modular discriminant delta and say e4 to the cube you can take the ratio and this object should have no automorphi factor at all so in fact it should be modular function not modular form it should transform without any automorphi factor and that's how it looks like and curious thing since I emphasize counting a lot this object also counts something its coefficients are also integers and it counts dimensions of representations of the monster group the maximal sporadic the largest sporadic group so there are lots of connections between this coefficients and counting of things but again they appear as random isolated facts and what I would love to see if there is an answer to this question can reconstruct them as index of something so finally for purposes of my talk I want to introduce one last ingredient which also has very nice story or character attached to it namely character of Ramanujan last year there was a very nice movie I encourage those of you who haven't seen it to go and see it it's also very touching story unfortunately just like Teni Yamara Ramanujan died very young in his early 30s and it's very sad story but it's also very touching story and another touching fact about it is that he introduced yet another class of modular objects which at that time he didn't even think about modularity he was thinking of Q-series expansion and Q-series properties so for example a power series like that summing over M with Q to various powers determined by M is one of the expressions that he wrote for from peace point of view and now this class of objects goes by the name of mock modular forms and again there are various interesting subcategories and these are modular objects of 21st century I probably should say even though they're quite old their properties and structure we're still about to discover so that's the last class I want to mention and then I want to mention interesting intriguing property of this class which actually has very old history Don Zagir calls it strange identity and it involves an object which is also a close cousin of this mock modular forms in fact a special case of it that they showed in the previous slide and what's strange about this identity is that the two sides of this identity are never well defined at the same time the left hand side written as a sum of 1 minus Q times 1 minus Q to square and so on is only defined if Q is a root of unity and in fact Don was inspired by talk that Maxime gave exactly 20 years ago it was 1997 so now we should celebrate 20th birthday of this expression and in fact Maxime actually was looking for analytic continuation of with Anurashitikhin tribe and parents so I'm sure that somewhere in the back of his mind it was closely related to theme of this entire discussion we have today that in particular the rest of my talk so this side of the equation is only defined as roots of unity so if you picture the unit disk so this the left hand side only makes sense where the sum terminates and this are precisely roots of unity points on the unit disk the right hand side on the other hand is nice Q series expansion in fact it's roughly speaking half derivative of the Dedekind eta function so coefficients chi are exactly the same as on the previous slide and if you roughly differentiate Dedekind eta function which was also theta function like this it didn't have n in front and the sum so if you differentiate Dedekind eta function with respect to tau you'll pull down n squared and that's what half derivative refers to if you pull down just n and not n squared by differentiating with respect to tau you get this sort of expression so it's very much like Dedekind eta sum where I stick n times the rest of the Q series and this is perfectly defined inside the unit disk so right hand side is defined inside the unit disk when absolute value of Q is strictly less than one and left hand side is defined on the unit disk but of course what relates this guy is the property that if you take Q to be so this is Q plane if you take Q from inside the unit disk and ask about the limit as you go to root of unity you recover the left hand side so that's what this identity means and that's It also works for formal expressions in any root of one and make a significant step. So it can make kind of I think also formal expansion near any root of one the left hand side and the right hand side as well. Yes exactly yes exactly yes yes yes yes but that's right yeah so the limits and their expansions coincide in the vicinity of root of unity so that's what the strange identity means and my goal in the second half of the talk is present to a context where first of all the strange identity is not strange at all where it's natural and it happens a lot and also answer the other question about role of integer coefficients. The function when it's very discontinuous on the on the disk or on the on the subject because it's only for fraction that you can make some sense of it. So the left hand side is only for fraction meaning on it's yeah it's very discontinuous it's only defined at roots of unity and yeah but this this is defined inside. So what happens if you reach any other point on the unit circle not a little bit? Well again it's it's pretty badly divergent I would say I don't know what happens because if it's not a root of unity. Maybe I don't know that's a good question at least I don't know that. Okay so second half of the talk is going to be completely different subject so and that's where like I say making connections and bridges is natural in fact the question about modular forms that I ask about interpreting that coefficients as index of something actually comes from this other field of mathematics that has to do with three manifolds and low dimensional topology. So there my question and I'll also pose this as a question is how to associate a Q-series to a three manifold. So that's that's my second question and I don't really know the answer so if you know anyway how you associate Q-series to a three manifold for this you probably would be low dimensional topologist please tell me because the answer is either going to be related to what I'm going to tell you next or it's going to be something new and something cool. And I would be really interested I'm not joking I'm really interested. Then in this field we can also ask analogous question if you have a Q-series just like we had in the previous first half of the talk it will be expansion with respect to Q if it happens to have integer coefficients it's also natural to ask and people in this field ask a lot this kind of question what are the vector spaces that have the graded vector spaces whose graded dimension these integer coefficients are encoded. So for from a viewpoint of low dimensional topology this is a starting point for asking more interesting question about vector spaces homology but first you have to have a Q-series with integer coefficients. And again I actually don't know how to associate Q-series to a three manifold or other objects in low dimensional topology so I would be extremely interested to hear. So one standard variant of three manifolds which is very powerful is Wittler-Russchitzelian porive variant it's computed by Chern-Simons path integral exactly as Marcus explained in his morning talk. And it's a path integral over gauge connection A whose exponent is K times Chern-Simons action as is the Chern-Simons functional that Marcus wrote on the blackboard and K is the level integer that has to be integer because the integral needs to be well defined. So in this story as well as in many other quantum field theories you can start Feynman diagram expansion which is basically saddle point expansion of the integral this is something that physicists do and love and that's precisely a good starting point for resurgence because you obtain perturbative expansion or a saddle point approximation to the integral. You have vastly growing number of diagrams in fact this is stolen from paper of Marcus where this table and examples of diagrams show you the number with different number of loops r is the number of loops maybe minus one and d is the space of such diagrams. So like I told you in the beginning both Marcus and Maxim influenced a lot of what's going on here so again I want to thank them for sharing that wisdom and hopefully not blaming me for stealing the pictures from the papers. So like Marcus said we can start doing perturbative expansion to this path integral and try to approach it with resurgence theory. So in this context we'll use two versions of resurgence theory one is roughly 200 years old goes to Picard-Levschitz talks and so on and unfortunately it will not be enough to treat some of the questions that arise in trans-seimant theory and its analytic continuation so we'll have to resort to resurgence of Picard-Levschitz theory which no pun intended is called resurgence. So that's only 20 plus something years old and again there are many key players here in France so this will be version 2.0 of Picard-Levschitz theory that will be able to track certain technical difficulties that we're going to see in this analysis. If you're naively applying Picard-Levschitz theory to Schrodinger and Simons path integral you expect roughly the following you expect that if you complexify your variables or fields in this case you naturally will get expansion where you have subtle points that are various flat connections and again if you complexify this will be complex flat connections on a three manifold M3 and for each one of them you have this kind of trans-serious expansion where you have exponential factor that I made very explicit is basically trans-seimant functional of a given flat connection times perturbative series expansion in 1 over k so for purposes of this talk I use several variables so q is exponential of h bar and this is same as 2 pi i divided by level k so the small parameter is roughly 1 over k. So that perturbative... So the L3 is real dimension... Yes this is three dimensional manifold critical points or subtle points of this action functional are precisely flat connections and M is the set of flat connections. But do you say complex how do you complexify? Because in resurgent analysis as well as in per-carolabschist theory natural thing is to complexify the original variables. So over the three manifold you have some... Right....connections but connections are real? Well originally we had g bundle over three manifold M3 and then naturally we passed to gc bundle basically meaning that if you had connection A you now which had some reality condition for example it was Hermitian, anti-Hermitian now you make it complex valued and declare that all complex values of its components are allowed so just formally complexify. Because doing this may help you find subtle points which are not necessarily on a real axis and they still may have important contributions. So you formally complexify everything you had before and then decompose... Right so I should probably not say. So there are two versions or two ways to look at such integrals using subtle point approximations. One is where given each critical point which I denote by alpha here you associate with all trajectories of steepest descent that Marcus mentioned earlier and they usually sweep out some surface called left shift symbol so that's a picture of a left shift symbol and then you can project this left shift symbol onto what will be called Borel plane. This is basically projection by your action functional so S of A projects you to complex plane so values of S is precisely what I'm going to call Borel plane later. So the thing is so then what you get is this naively you would expect that your with-narration to arrive in Borel in this case has expansion of this form where this is perturbative series around each subtle point some over subtle points with some coefficients and alpha which are related to what Marcus called C and determine the choice of the contour. The problem is and this is one interesting thing where picor left shifts is not going to be enough is that in terms of Simon's theory each flat connection has infinitely many lifts and that's a delicate thing. So the space of all gauge connections, modular gauge transformations is not simply connected. You may go on a closed loop kind of like this and each alpha has infinitely many copies. That's related to the fact that in four dimensions we have instanton number and that's also related to the fact that transimons of alpha is the same as transimons of alpha plus any integer namely the transimons function itself is only defined mod integers. So what happens then is that instead of one subtle point you have infinitely many towers in infinite towers of subtle points. So that's already kind of tricky and that's why I introduced here this notation bold alpha which denotes the lift of your subtle point into this tower. This is interesting thing and I think I should have interesting applications to other problems of trigonometric or hyperbolic nature. So if you have two different equations, if you have trigonometric or hyperbolic integrable system you'll have similar phenomenon. You'll have sometimes this infinitely many lifts of a single subtle point. If your equations are written in trigonometric form. So that's cross that means that now we are working on the universal cover. So instead of space of all gauge connections I have to introduce the universal cover and Z tells me on which sheet of the universal cover I am. Because you have second to next alpha theta on the first formula and alpha zero on the second on the theta zero. Right. So theta will appear in a second and this bold phase alpha is again the lift. So this basically says that if you had, if you thought that your manifold had say three flat connections now what I'm telling you is that it has to have infinitely many copies but ends should be such contributions of all of them, of all this towers should be such that each individual and alpha and bold phase alpha sum up into what we used to call n alpha. So n alpha is contribution of this connection alpha and bold phase alpha means it's integral lift. So that's first delicate point that you encounter again on gauge theory trigonometric hyperbolic integrable systems things like that. Q difference equation. But your gauge group is here, is here to see and you're descending down to SU2. Here I was saying that, okay sorry that's a good question indeed. So here this last part of the equation says that originally this written Rache-Sichin drive invariant is defined by SU2 gauge theory. So our contour is such that all real points should contribute with coefficient one. At least that's what we, well that's basically part of the choice of the constant C that Marcus was telling us in the first part of the talk. So it's additional data, it basically tells you how to set up your problem. And the problem is set up in such way that you want the coefficients to be one if alpha is SU2 flat connection and zero if it's not. So that's the definition if you wish. It's part of the definition of how we want to define trans-heimance path integral. We want to define it by integrating over real slice in this complexified space of fields. So that's basically the second. It is not really well defined from this point up here. Exactly, and yeah, that's what I believe I say on the next slide. Indeed it's not well defined and this left-shift symbols exhibit the car left-shift monodrome is. So if you start there in parameters such as H bar or Q or other parameters you will have stocks phenomena and stocks phenomena are basically saying that if you go from one configuration to the other where I just draw for you projection on the Burrell plane what happens is that a given contour gamma receives contributions from other contours and the intersection numbers will determine for you what's the coefficient. So that's another interesting phenomenon I want to emphasize. In a finite dimensional problem the stocks coefficients or at least in Picard left-shift theory these coefficients are integers because they're intersection numbers of some cycles left-shift symbol and left-shift anti-symbol. So what happens? There may be zero cycles and you can you are integrating over the zero. For the cycles which gives you zero. So you must specify the cycles. In usual Picard left-shift theory these cycles are mid-dimensional in the space of complex fields. So their intersection number is again if you're in a nice situation it's just a number. It's an integer because it counts the intersection points of two mid-dimensional cycles. So it's just an integer. But I want to emphasize that here because of the first issue that we had infinitely many settled points what's going to happen is that effectively this number will become fractional and that's a pretty cool phenomenon because we'll have some of integers some of say plus minus ones and they will be they'll need to be regularized resummed and as a result you get some interesting fractions. So this is another tricky point which usual Picard left-shift theory cannot handle but resurgence does extremely well if it's very clever. So that's I think where I'm heading next. So you want to treat this kind of problem using resurgent analysis which projects everything to the Borel plan. You don't have to think about this infinitely many copies in the space of fields it will do it for you by itself even if you are very naive and but if you care about what happens in the space of all the variables you can do that too but it will take care of it. And very naively what you do is the following you take your perturbative series such as this H bar expansion you divide by r factorial which are this power loop coefficients exactly as Marcus explained in his talk and produce Borel sum BZ of psi and then just like a number theory you ask about analytic structure of this function. You don't just do the summation you ask does this have analytic continuation to complex plane and if so what are the singularities what are the residues very much like what we did for L functions. Now let's see so this picture is illustration of Borel resumption where you only sum not up to infinity but just take first couple of terms and this is what you can easily do with Mathematica if you're like me and don't want to spend too much time on either thinking about abstract argument or even in Mathematica you don't have to have high power Marcus is extremely good at this I know only how to do say five terms or three terms it's quite amazing that this actually is very nicely behaved and pretty fast convergent at least in good examples so this is done with very poor accuracy but you already can start seeing that something is probably happening on imaginary axis that you may have some singularities this is supposed to be profile plot for for this Borel sum. Indeed what happens you have poles on the imaginary axis and again I want to emphasize with the slide that what you have to care about to keep in mind is what are the singularities on the Borel plane and what are the residues that basically at least for me tells what resurgence is supposed to produce and encodes the most essential data about the problem so it's like a snapshot like a photograph of Borel plane that that that says a lot. So in this case again coming back to our example you perform lateral Borel summation and because everything was on a all the singularities on the imaginary axis you can deform contour and basically some of their residues but the only thing that's going to happen is you're summing over infinitely many sedals so in the Carl Laphs theory it's very hard to develop the homology theory or any sort of homological machinery which deals with this infinite number of intersections and so on but in the Borel plane that's very easy we just have infinite sum of residues and that's not a big deal and it produces for you some final answer in the form again that Marcus explained and here I think this quote by Hallemard is very appropriate especially in this occasion and right so the two key ingredients are the coefficients and alpha beta which tell us how corrections to saddle point alpha will come from other saddle points these are called trans serious coefficients and then position of singularities on the Borel plane determines for you the exponentials this is precisely what Marcus explained in his introductory talk. So since he already spent quite a bit of time on exponentials I'll emphasize for you the structure of this trans serious coefficients and alpha beta as I already mentioned they come at least in this kind of problem or in any other trigonometric or hyperbolic problem from infinite sums and you get sums of plus minus ones which are basically the intersection numbers only thing you have to sum infinitely many times plus minus ones you can do it using zeta function regularization you get this kind of expression which essentially then boils down to some fractions okay and that's a very interesting phenomenon which in this context leads to a miracle that I'm going to summarize for you in the form of a theorem so this theorem comes from paper of myself Marcus and Pavel and it says something interesting so if you have in the turn time is there you have different flat connections which have different stabilizers so part of the gauge group may still fix your flat connection and if that happens the flat connection is called reducible so it turns out if you think about it from the viewpoint of the Corleaves theory or resurgence theory that reducible flat connections are such that they can receive contribution from non-nibelian ones as trans series but not the other way around so there is directionality so irreducible and reducible are not the same at all and again part of this has to do with the fact that they have stabilizers so that's an interesting theorem and the consequence of this theorem is something cool it says that you can decompose your Witten-Rischy-Tiching-Turivon variant into some of blocks call them basic classes for instance by analogy with Donaldson theory how it decomposes into cyberquidon variants basic blocks or basic classes labeled just by Ibelian flat connections and it turns out that these guys do have nice analytic continuation inside the unit disk even though original fellow doesn't again I have no idea how to analytically continue Witten-Rischy-Tiching-Turivon variant from roots of unity. Ibelian means reducible. Yes sorry I'm talking about SU2 group so for me reducible Ibelian are the same but in general you are absolutely right reducible so Witten-Rischy-Tiching-Turivon variant is only defined at roots of unity and a big question in this field is how to give a Q-series again I don't know the answer to this question but what I do know is that to a given three manifold you can associate not just one but collection of Q-series expansions labeled by Ibelian or reducible connections each one is well defined in the unit disk and they do have exactly this property that if you approach roots of unity from inside the unit disk you get the corresponding value. All the complex elements are away right these are all unitary connections. These are unitary connections? No no no this is complex. This is complex with a point that if Q approaches root of unity they have to reproduce Witten-Rischy-Tiching-Turivon which is the real slice. I meant the connections are unitary connections or complex in this discussion? In principle if you are inside the unit disk if Q is complex any Q with less than one norm then all flat connections real and complex contribute to this expression so all of that. But from the point of view of partition function we are calculating partition function of chance simmers on M3 of unitary chance simmers or we are trying to calculate the sum. But then it decomposes into sum of all possible contributions. The function is of unitary connection but it is rewritten in terms of something which is a billionized and complex. And it is a billionized in a very funny way so again it is a billionized in a way where you take so maybe that is another question I wanted to raise where you have. I can't think about Z with a subscript a M3 as some kind of a billion partition function. You can but I don't know how to do it in terms of simmers theory. Using some string theory language you can. So I feel that this may be getting a little abstract let me see a show of hands I am going to finish in five minutes don't worry. How many of you are happy with Picor Lefcher's theory just to make sure that I am not losing everybody? Just Picor Lefcher's theory. How many of you are happy with Picor Lefcher's theory with infinitely many Picor Lefcher's monodrome is happening at the same time? Okay not so bad. So for the rest of you that is the luxury of the physics talk. I have a demonstration so I just want to explain what is going on here. So we have a theory where we have infinitely many critical points, subtle points and let me represent each subtle point. So by bin. So this is going to be reducible flat connection. This is going to be irreducible flat connection and that's another reducible guy. So to make it easier I will denote irreducible ones non-abilian flat connections by red crystal balls. There are infinitely many copies of each flat connection each subtle point because of this lift to universal cover. So I represent them by infinitely many stones, red stones sitting here. So that's going to be reducible. So middle one will be irreducible. You have infinitely many copies because of this infinite cover issue or hyperbolic nature of the problem of a subtle point. And this is going to be another non-abilian flat connection. So reds are abelian, white is non-abilian. So what happens in this process of Borel resumption or resurgence is that this guy gets distributed among the abelian ones. So the contribution is still there just like it was in the Witten-Risch-Tichen-Turibund brand. Witten-Risch-Tichen-Turibund brand sums all of them with a coefficient one. But what happens here is that first of all you have infinitely many copies. That's the first phenomenon. And second is that because of the Picorolevschitz monodromes that a plus minus one summed infinitely many times, this guy, non-abilian fellow, gets distributed with coefficients say one third and two thirds. So what I should do to produce this object, ZA, on the right hand side, I should redistribute. So these are now ZAs. And they do have everything we had before where total now is divided just into reducible fellows. So hopefully this will clarify for you what Picorolevschitz theory with infinitely many leftist symbols means. And maybe as a question, and I think I should probably conclude here, I want to ask for other phenomena maybe in number theory where the following happens. You have some entity which is total, and for us this was total Witten-Risch-Tichen-Turibund brand. And in my example it was contribution of reducible, irreducible plus another reducible. And the middle guy gets redistributed into these two with some coefficients say one third, two thirds such that the total sum is still there. And I want to ask if you know of any other phenomena, for example in number theory or elsewhere, where some objects of slightly different nature, there should be some analog of reducible, irreducible, maybe a billion, non-a billion, get redistributed in very similar way. For example, I could find L packets in the Langlands correspondence being somewhat analogous where again you have certain representations and you package them together. You group them in classes with certain coefficients which are kind of analogous to this trans-series coefficients. But I would love to hear more examples. So finally, I don't want to take more of your time. I want to point out that once you start with this perturbative series expansion in 1 over K in trans-simon theory, you do this clever Borel-Ris summation. Resurgence is very smart. It takes care of all this pole summations and so on. You produce this object ZA and these guys now have, first of all, by definition because we were summing over steepest descent contours, they're well-defined inside the unit disk. So this is actually Q-series. And second of all, they turn out to have integer coefficients. So some magic happens where for any three manifold in this trans-simon theory, not only you continue your written Rishi-Tikhin-Turban variant away from the roots of unity inside the unit disk, for each of these basic blocks, ZA, they start having integer coefficients. As functions of Q. So that's, to me, magic. I have no explanation of this from a mathematical point of view. Physics suggests why it should be the case. But that's at least the result. So it turns out that these are precisely mock modular forms that we saw on the previous slide, at least in the context of Cyphert manifolds. Probably not in general, but at least there. Okay, I think my time is up, so I should stop. Thank you for this next talk. Other questions? ZA, do not have integer coefficients. So ZA are basically, yeah, so there is a question of notation in our paper. There are ZAs and ZA hats, and there are simple linear combinations of each other. So I promised you two theorems, and I managed in my time to get only to the first. One theorem would tell you which linear combinations have integrality. But that's, I would say, a slightly simpler issue. Yeah. Yeah. Regarding the categorification of Q series, the first question. So as far as I recall, there is a paper with specter. He actually used this written index and derived the function as some measure of some physical system, something. So I think this kind of configuration exists in physics literature. Then I think Allen Combs and Barnall, number theorist. So he also has similar things, but derived from the specter's paper. The specter of two papers, one is half-leap broken on half-leap broken supersymmetry. And what is the Q series network? I think he constructed something out of that, since I don't recall. For some reason, information is just very common. Well, I would love to have the reference. And again, for some isolated things, of course, this is easier. My question is how general would that be? So here, for example, this is for class of mock modular forms. I would love to have it for some big classes, not just one single example. But anyway, I'll take over. But this is for the zeta function. It's somehow related to some green Hilbert-Fuller construction. So, uh-huh. OK, sounds good. Thank you. Very, very useful. So, explain why the theorem one is true. OK, so theorem one is true for the following reason. And again, that should be also a good lesson for resurgence. So what happens is you have subtle points, one, and irreducible guys are non-degenerate subtle points. Subdual ones are very degenerate because you have further action by stabilizer of the group. So they're actually steki. So what we're dealing with, we're dealing with Picard-Levschitz monodrome is, or crossing types of singularities were. A1, so to say. Huh? So you're saying it's not A1. It's not A1, no, not at all. Right, so from a singularity theory point of view, and in terms of assignments, this is unavoidable because trivial flat connection A equals zero is always a flat connection. And it's highly reducible. It's all of the group G stabilizes that guy. So in trans-simon theory, you cannot have a situation where all your critical points are nice, more, so even more is bought. You're in a situation where there is this big degeneracy. And what's happening is basically the denominator, the quotient by additional stabilizer shifts the Mohr's index in such a way that Levschitz symbols, if you think about it from Picard-Levschitz monodrome point of view, don't even see each other. So they're in a different grading or different dimension. So from Mohr's theory point of view, they're in their own position, not mid-dimensional anymore. So that's the difference between reducible and reducible. So as a smoking gun for a surgeon's or quantum field theory or anything else that you may be interested in, if your singularities are not just poles or not just simple singularities in Picard-Levschitz theory, there is some degeneracy, you should expect phenomena like this where there will be directionality. So in this case, again, reducible can pick up irreducible as pieces but not the other way around. It's all about singularities. Is it independent of M3, right? Yeah, yeah, yeah. It's independent of M3 and in fact it's independent of this infinitely many copies. It's just about behavior of Picard-Levschitz thingables for degenerate subtle points. So and unfortunately or unfortunately, they're completely unavoidable in trans-heimance theory so we had to deal with that. You have one more question? So at the beginning of the talk, you asked question one about some commercial origin of as a stensiles. You know there is a stensile symbol that gives you as a stensiles of the form that you wrote. The lattice sum typically comes with a character in the numerator and it's associated with cycles. So for example, you can get a version of the E4 series with some character from three cycles on the Kligow varieties. It's called Beninson construction and from that you can lift up by competing some asian or something that is induced by the homology or the homology of the main. I see. So there is some HMM, some doubly-graded homology groups that give you as a stensiles here? So there is something that is attached to the cycles that is called the as a stensile symbol that leads to some as a stensile with a homomorphic one with characters and they are just put in place by Beninson sometime ago. Well excellent, I would love to learn more about it. Okay, so I hope that we stop now and we do in ten minutes. Thank you.
 Well, first I want to thank the organizers for organizing this meeting and in fact running  it for a number of years as well as for inviting me here.  I have to say that I am myself really new to resurgence at least compared to all the  other speakers.  In fact, 95% of what I'm going to tell you I learned from either Markus or Maxim and  I want to thank them for sharing their wisdom, so that was very helpful.  The only little piece which is new will consist of two concrete theorems that I mentioned toward  the end of the talk and these two theorems come from two papers, one of which is with  Markus and he's very bright student Pavel Putrov and another is from paper of Pavel  and Kumrun Bafa.  So in this talk I'll try to make connections between two completely different subjects,  one which already appeared earlier this morning and the other one which has to do with modular  forms.  So I'll try to show and illustrate how we can go back and forth between different subjects.  First of all by asking questions of one subject in the context of another, how the links sometimes  can provide the answers to questions, sometimes quite surprising answers which are not expected  if you're narrowly focused just on one particular topic.  It's a good illustration that bridging two fields may actually help to understand structure  of things that we're trying to study.  And I'll also raise a lot of new questions by going back and forth and by thinking about  this in a broader context will lead me to lots of questions that I'll now use as opportunity  to ask during this meeting.  In fact if any of you happen to know the answers please get back to me and I'll be happy to  learn more.  So like I said I'm new to this and I'll enjoy learning this even further.  So I'll start with theory of modular forms and this will be somewhat pedagogical.  I'll soon get to the place where I want to be but I'll try to frame it in a more canonical  context so that if you're interested just in resurgence and have less familiarity with  modular forms this won't be too harsh on you.  So modular objects or modular forms specifically are defined as objects that transform nicely  in this case under SL2Z modular group.  So F is our function of variable tau and we require that as if we perform fractional linear  transformation on tau F transforms in a nice way.  Here a nice way means that it gets multiplied by C tau plus D to the power k.  K is usually called the weight of the modular form and this factor is an automorphic factor  that again appears in theory of classical modular forms.  Later in the talk I'll show you somewhat more abstract or different versions of modular  transformations where you can modify this rule in various ways.  For example you can add additional terms and so on so depending on how you set up your  story you'll get slightly different variants of modular objects called forms and used with  other adjectives but this is the standard one.  So let's continue from here.  So our starting point is a function of variable tau and we want to characterize it.  I also should say that here I'm focusing on standard Mobius group which is SL2Z.  You can also look at various finite index subgroups or other modular groups.  So at least for SL2Z the story is generated by two generators conventionally called S and  T represented in matrix form as shown on the slide and one of them basically inverts tau  is asking how function of tau is related to function of minus 1 over tau.  That's interesting.  That's in physics would be called strong recoupling duality but T does something a little simpler.  It basically says that function of tau plus 1 is the same as function of tau and there  is no contribution from this automorphi factor in front.  So this second fact that f of tau plus 1 is f of tau immediately tells us that it's a  good idea to first of all, expand and shape tau, introduce Q which is exponential of 2  pi i times tau and then this second rule will be implemented automatically.  Also shown here is the fundamental domain and various translates our fundamental domain  with respect to SL2Z elements.  So for example shift by plus 1 takes fundamental domain here by minus 1 here and S does the  inversion so we have this form of the cusp instead of going to infinity and so on.  So this is how standard theory of modular forms usually begins and just like I mentioned  a moment ago the T generator that tau is supposed to be same as tau plus 1 tells us that we  should we better work with variable Q which is e to the pi i tau and in the rest of this  talk it is this variable Q which will be playing the central role or in other words I'll be  asking all kinds of questions how Q is related to various other types of variables that will  appear and we'll be thinking about function of tau actually as a Q series as a power series  expansion on Q with some coefficients a n which are priori are completely unconstrained  they can be anything as long as this object satisfies this modularity requirement that  under S transform or more general SL2Z transforms it behaves in a way we wanted to behave.  So that's where it starts so again if you open a textbook and start learning this subject  from scratch you may ask what are the simple examples of modular forms let's try to construct  them and standard way to construct something which behaves nicely under the group is roughly  speaking to sum over images with respect to the group or perhaps sum over lattice points.  So then you can pick different entity which you're trying to sum over lattice points in  this case say lattice is of the form of complex numbers which are of the form m plus m tau  where tau is complex variable m and n are integers and you can try to sum image of the  rational function or it's a translate set that there's different points which basically  gives you this Eisenstein series so for each given k again remember k is the weight this  defines Eisenstein series and you can nicely write it in terms of Q, Q expansion where  sigmas are divisor functions these are sums of divisors to suitable powers for example  sigma 2k minus 1 of n takes divisors of n positive divisors of n raises them to the power 2k  minus 1 and adds them together.  So this is our first modular form or rather simple class of modular forms which you can  massage a little bit further you can do Lambert series resumption so on the right hand side  we saw that in the Q expansion coefficients are given by divisor function this sigma of  n coefficient of Q of n is sigma of n so standard Lambert series resummation says that that  also can be written in a much nicer form by introducing denominators which are also functions  of Q in fact in this you kind of get geometric series progressions which in the end resummon  to this divisor functions.  So the first non-trivial example is Eisenstein series E4 first of all we can quickly show  that for odd values of the subscript you get zero because you're summing something and  there are contributions with pluses and minuses so they cancel so first of all the subscript  has to be even and then E2 is a very weird guy you may want from the start even before  you start any analysis look at values of the subscript of this index greater than 2 simply  for convergence reasons because you're summing over two dimensional letters and of some of  a rational function whose denominators controlled by this index K so from the very beginning  you may want to look at E4 as your first starting point and again just as we discussed in the  previous slide this is how it looks like you can use this Lambert series resummation to  write it in very nice form where in the numerator you have n cubed Q to the n and then you divide  by this one minus Qn factors so after spending some time working with Don Zagir I learned  my lesson that if you ever have to write a Q series or any function it's always a good  idea to write first few terms so from now on in my life every time I write any function  or any series I always present first couple of terms.  You can't have negative powers of Q? No, no, no, never, at least in this subject so sometimes  you may want to require the leading term the constant term to be zero in that case you  get cusp modular forms and again you may impose further additional conditions which  basically tell you how it behaves at the cusp that's very interesting that constrains the  structure or modifies the structure a little bit but you definitely prefer not to go too  far in a negative region at least in the traditional theory of modular forms.  So this first non-trivial example of Heisenstein series E4 is quite nice and interesting again  here is the first Q term so it's Q series expansion and like I mentioned earlier you  can obtain modular objects by summing over lettuce and the Heisenstein series you can  obtain by summing over lettuce in several different ways.  One is by summing rational function with power 2k in the denominator that's how we introduced  it.  Another one is actually as a theta function where you sum exponential terms over lettuce  point and here in this case if you sum over root lettuce of E8 you get a theta function  which is in fact the same Heisenstein series.  So this can be obtained in a number of different ways and has many interesting properties so  this is very concretely if you haven't seen modular forms in action before that's how  it looks like.  Here is the next one E6 as I say only even values of the index or argument give you something  non-trivial so lembert reclamation gave us that formula again following Don's advice  I write for you first Q coefficients explicitly and here below in the plot I show how the  real part of this function behaves it's very nice inside the unit disk and it has interesting  behavior as we approach the boundary of the unit disk and later in the talk this phenomenon  of approaching the boundary of the unit disk will play an important role so not exactly  the way it's shown in the picture but in a closely related way.  So that's our first two examples.  We could try to go further and look at E8 and so on but one interesting fact about modular  forms at least traditional modular forms and presenting for you is that the condition  that we imposed from the beginning that object F under modular transformation is some factor  times F of tau is multiplicative and as a result if you have two objects which transform  very nicely with respect to modular group you can multiply them so they form a ring  and in this case it turns out that the two guys that already showed for you E4 and E6  in fact generate the ring so the ring of modular forms is freely generated by these two fellows  which means if I now start computing E8 it will be expressed as E4 squared E10 will be  basically E4 times E6 and so on I have to get to weight 12 until I start seeing something  more than one dimensional space of such objects.  By the way did you notice something else about examples I just showed you?  Integers.  Exactly.  So their coefficients are integer and this is kind of interesting because the original  way we define modular form did not require this integrality.  All we had is a requirement that it was a power series in Q that already used the T generator  and then this power series in Q should have had nice property with respect to S generator  which inverts tau but that did not a priori had to give anything with integer coefficients.  So it's rather peculiar that the answer turns out and in fact in many cases in the theory  of all kinds of modular forms the coefficients come out to be integers so natural question  to ask is what is the nature of this integer coefficients?  Are they counting something?  Actually in mathematics every time you see an integer number it's counting something  so the natural question is what is it counting in our case?  A more modern version of the question is if you see a number integer number you can ask  does it represent dimension of some vector space or maybe some cohomology theory?  And in fact already from our discussion we see some hints for it that our integers are  not necessarily positive or negative even the two examples I showed you indicate that  there is no positivity condition so they can be just dimensions of something or counting  something our numbers even if they have such interpretation are likely to be of the form  of an index where we sum various dimensions with plus minus coefficients.  So therefore what could possibly be is and that's a big question mark whether our Q  series expansion can be written as a graded earlier characteristic of some vector spaces  which are doubly graded and determines the sign and n determines the power of Q.  So as I mentioned earlier in this talk I'll try to ask lots of questions to which I would  love to know answers and maybe on this board I'll summarize my questions.  These are real questions for you and I don't want to go astray and use much time of the  talk for this but especially after please tell me if you have really good answers or  good ideas and my first question is even in the context of modular form is there a cohomology  theory which is doubly graded which provides which makes this a quality true?  So the fancy name for this finding such spaces is called categorification.  Anywhere in life where you see a Q series and if it has integer coefficients it's a  good habit to ask this kind of question and again modular forms provide lots of examples  of Q expansion with integer coefficients so therefore I want to ask is there a cohomology  theory which really makes it true?  In some cases it could be realized as not quite with an index but elliptic genus so  it is an index but again I don't really know.  Again if you happen to know the answer please tell me I would love to hear I don't know  I honestly I don't know.  In mathematics the closest I could find is et alchoromology but that's not exactly answering  this question for example I cannot just say that H's there are et alchoromology groups  unfortunately not so if again you know the real version of this answer please let me  know I would be very curious and again I will try to ask analogous questions as we go.  So with this frame of mind what are the integer coefficients counting can we realize them as  graded dimensions of cohomology groups of actor spaces?  Let's continue and in one class of examples of modular forms I will give you affirmative  answer where the answer will be crystal clear and in fact it's not just exists it's actually  useful for people in completely different area so in that case there will be answer  and that makes me hopeful that more generally that can be true.  So continuing with the crash course on modular forms we should go a little bit further and  like I mentioned to you earlier the next interesting thing appears if we start going increasing  weight of modular form in weight 12.  So first we introduce dedicated eta function which is given by that earlier type product  of 1 minus q to the n and this function by itself is already a pretty cool function.  First of all it has q-series expansion as a sum and it has this theta like expansion  with coefficients which I denote chi n these are characters of something of conductor 12  that is not terribly important but the same coefficients will appear later in one of the  following slides but already this function itself is counting something cool if you take  that infinite product forget about the pre-factor q to the 1 over 24 and just invert it then  you see that you have 1 divided by bunch of geometric series progressions or factors which  can be expanded in geometric series and that expansion will have positive coefficients  and those positive coefficients in q expansion if you invert this infinite product is actually  counting two-dimensional Young-Tablos or 2D partitions which plays very important role  in many subjects from random matrix theory to modern gauge theory.  Now in a course of modular forms dedicated eta function is interesting but what's even  more interesting is its power 24 which gives you function delta called the modular discriminant  and this is modular form of weight 12 this is not so hard to show by taking logarithmic  derivative of delta of tau and like I told you before everything should be generated  by our ring of modular forms is generated by e4 and e6 so delta should be expressible  in terms of e4 and e6 in fact they're exactly two non-trivial well the space of weight 12  modular forms is two-dimensional so therefore everything should be combination of e4 cubed  and e6 squared and delta is such a combination given by this expression here so you can  multiply it out to get the first two terms as Don Seguir taught me to do for every function  so to understand the meaning of this integer coefficients there is actually an answer in  theory of classical form modular forms of course but it's not in the form I wanted it's  not in the form of this categorification where you take coefficient and write it as graded  holi characteristic it's much more contrived and not satisfying for our purposes but there  is an answer and as an honest guy I'll tell you what the answer is.  So Don Seguir goes via Borrell transform and if I were in Mil Borrell I'll definitely  write reference complaint it could go something like this hey there is in a closely related  context a very similar transform called Borrell transform that you saw earlier this morning  which basically does this so given a function f of x you can produce it's in this case a  number theory usually called Mil in transform for whatever reason transform which is a function  of variable s and I want to point out that it is indeed closely related to Borrell transform  of earlier this talk and much of what's what you're going to see during this day if you  write x in an exponentiated form if x is exponential of some other variable then basically  what this does is precisely the Borrell transform which you see in resurgence a lot of time so  that's already a very interesting nice connection to the Borrell transform and integral also  goes over ray in this case goes from zero to infinity and so on.  So we apply this version of the transform Mil in transform to our modular form to obtain  a function which again in the context of resurgence would be some version of Borrell transform  that encodes some information about integrality or gives partial answer to this integrality  that I wanted and what we get is a function and a number theory just like in resurgence  it's a good habit to ask about analytic properties of this function it's a natural to ask a question  like where does it extend in a complex plane where the singularities are what are the poles  what are the zeros residues and so on.  So this transform in this case specifically gives L function and there are several versions  of L function one I denote by lambda and the other by L related by this gamma factor.  Lambda is more natural if you want to ask analytic questions about extending this to  complex plane and then in fact if you apply this Mil in transform to our class of modular  forms or closely related objects that we introduced before you're going to find that  it's an entire function and has two poles one at s equals zero the other is at s equals  k where k is the weight it usually has some symmetry where you exchange the two and the  coefficient of this the residue of this poles the coefficient vanishes if the leading term  in the q-series expansion vanishes for example for cusp forms this would really be entire  function.  So for analytic number theory this lambda is more natural for algebraic number theory  where we ask about integrality this L object defined by a prefactor gamma is more natural  L is the Dirichlet L function of a modular form a modular object F and you can define  it more directly simply by taking coefficients of the q expansion of our original modular  form which was sum over n a and q to the power n and dividing them by n to the s so formally  this defines some function of s and then again you ask same same type of question what kind  of function is it what is it analytic structure that I briefly mentioned in the previous slide  and so on.  So in number theory there is an answer to to to integrality of these coefficients and  namely they're counting something they're counting points on varieties over finite fields  so first of all you take that L function you write it as product over all primes up to  some finite set which may be related to a discriminant of your variety or maybe empty  set in fact and you get something like this you have universal factor for for each prime  and again weight k appears over there and when k is equal to 2 in a special case of  way 2 what happens is that the number of points sorry this coefficients that appear there  this AP's these are basically the same coefficients of our original series except that we're taking  them at prime value of the index n they're counting number of points on elliptic curves  over fields finite fields fp with p elements so AP is often called p defect because naturally  you would expect that roughly speaking number of points would be same as number of points  in this field plus one which is p plus one and a describes how much your answer is really  different so the integrality of coefficients a at least for prime values of p and then  multiplicatively for others are related to numbers of integer points over elliptic curves  so this is a classical result which originally was a shimura to neyam a conjecture and then  rediscovered by entree bay and finally made into mathematical theorem by andrew wiles  and many of his collaborators and followers so this is pretty beautiful subject and it's  interesting that the people involved had very interesting character alive so each of this  guys is a quite interesting character unfortunately ten yama died very young so he committed suicide  I didn't know this but preparing for the talk I checked the history of this on wikipedia  and discovered that actually his fiance also committed suicide shortly after and that's  actually very touching story of true love so that's that's quite interesting Gora Shamura  is also interesting character and at keltec we had visiting committee earlier this week  before I came here and digress told me about interesting discovery that Gora Shamura made  in princeton so he spent much of his life and how big part of his life in princeton and  princeton math building is very boring it's like a tower with many floors about 10 or  15 of them and it's it's uncharacteristic building all floors identical it's it's it's  a very boring looking structure so princeton students made a very nice prank one day so  they basically switched buttons one two three four and so on in the elevator so if you press  button seven you would go to fifth floor for example so what else they did they also switched  the office labels accordingly so you come out of the elevator and you go to all holes  exactly identical so you go to your office and the office in the same place it has your  label but then your key doesn't work so this morning security on campus got lots of calls  from people that their keys do not work and Gora Shamura was the one to make a discovery  that actually the levels were different simply because he decided to leave his office and  take the stairs down and then he suddenly realizes that he has to go many flights many  more than he would normally do so he made many interesting discoveries if you so here  the while's theorem formally known as shimurritin yamakonjecture is about weight two modular  forms which makes a correspondence to elliptic curves over finite fields but in in higher  weight and in other variants of modular forms this is part of modularity program where you  are not necessarily counting integer points of the full variety itself but of its certain  piece called motif and that's an ongoing interesting research program so continuing with our crash  course on modular forms in weight 12 as I told you before there are only two non-trivial  generators so if you already have our modular discriminant delta and say e4 to the cube you  can take the ratio and this object should have no automorphi factor at all so in fact  it should be modular function not modular form it should transform without any automorphi  factor and that's how it looks like and curious thing since I emphasize counting a lot this  object also counts something its coefficients are also integers and it counts dimensions  of representations of the monster group the maximal sporadic the largest sporadic group  so there are lots of connections between this coefficients and counting of things but again  they appear as random isolated facts and what I would love to see if there is an answer to  this question can reconstruct them as index of something so finally for purposes of my  talk I want to introduce one last ingredient which also has very nice story or character  attached to it namely character of Ramanujan last year there was a very nice movie I encourage  those of you who haven't seen it to go and see it it's also very touching story unfortunately  just like Teni Yamara Ramanujan died very young in his early 30s and it's very sad  story but it's also very touching story and another touching fact about it is that he  introduced yet another class of modular objects which at that time he didn't even think about  modularity he was thinking of Q-series expansion and Q-series properties so for example  a power series like that summing over M with Q to various powers determined by M is one  of the expressions that he wrote for from peace point of view and now this class of  objects goes by the name of mock modular forms and again there are various interesting  subcategories and these are modular objects of 21st century I probably should say even  though they're quite old their properties and structure we're still about to discover  so that's the last class I want to mention and then I want to mention interesting intriguing  property of this class which actually has very old history Don Zagir calls it strange  identity and it involves an object which is also a close cousin of this mock modular forms  in fact a special case of it that they showed in the previous slide and what's strange about  this identity is that the two sides of this identity are never well defined at the same  time the left hand side written as a sum of 1 minus Q times 1 minus Q to square and so on  is only defined if Q is a root of unity and in fact Don was inspired by talk that Maxime  gave exactly 20 years ago it was 1997 so now we should celebrate 20th birthday of this  expression and in fact Maxime actually was looking for analytic continuation of with  Anurashitikhin tribe and parents so I'm sure that somewhere in the back of his mind it  was closely related to theme of this entire discussion we have today that in particular  the rest of my talk so this side of the equation is only defined as roots of unity so if you  picture the unit disk so this the left hand side only makes sense where the sum terminates  and this are precisely roots of unity points on the unit disk the right hand side on the  other hand is nice Q series expansion in fact it's roughly speaking half derivative of the  Dedekind eta function so coefficients chi are exactly the same as on the previous slide  and if you roughly differentiate Dedekind eta function which was also theta function  like this it didn't have n in front and the sum so if you differentiate Dedekind eta function  with respect to tau you'll pull down n squared and that's what half derivative refers to  if you pull down just n and not n squared by differentiating with respect to tau you  get this sort of expression so it's very much like Dedekind eta sum where I stick n times  the rest of the Q series and this is perfectly defined inside the unit disk so right hand  side is defined inside the unit disk when absolute value of Q is strictly less than  one and left hand side is defined on the unit disk but of course what relates this guy is  the property that if you take Q to be so this is Q plane if you take Q from inside the unit  disk and ask about the limit as you go to root of unity you recover the left hand side  so that's what this identity means and that's  It also works for formal expressions in any root of one and make a significant step.  So it can make kind of I think also formal expansion near any root of one the left hand  side and the right hand side as well.  Yes exactly yes exactly yes yes yes yes but that's right yeah so the limits and their  expansions coincide in the vicinity of root of unity so that's what the strange identity  means and my goal in the second half of the talk is present to a context where first of  all the strange identity is not strange at all where it's natural and it happens a lot  and also answer the other question about role of integer coefficients.  The function when it's very discontinuous on the on the disk or on the on the subject  because it's only for fraction that you can make some sense of it.  So the left hand side is only for fraction meaning on it's yeah it's very discontinuous  it's only defined at roots of unity and yeah but this this is defined inside.  So what happens if you reach any other point on the unit circle not a little bit?  Well again it's it's pretty badly divergent I would say I don't know what happens because  if it's not a root of unity.  Maybe I don't know that's a good question at least I don't know that.  Okay so second half of the talk is going to be completely different subject so and that's  where like I say making connections and bridges is natural in fact the question about modular  forms that I ask about interpreting that coefficients as index of something actually  comes from this other field of mathematics that has to do with three manifolds and low  dimensional topology.  So there my question and I'll also pose this as a question is how to associate a Q-series  to a three manifold.  So that's that's my second question and I don't really know the answer so if you know  anyway how you associate Q-series to a three manifold for this you probably would be low  dimensional topologist please tell me because the answer is either going to be related to  what I'm going to tell you next or it's going to be something new and something cool.  And I would be really interested I'm not joking I'm really interested.  Then in this field we can also ask analogous question if you have a Q-series just like  we had in the previous first half of the talk it will be expansion with respect to Q if  it happens to have integer coefficients it's also natural to ask and people in this field  ask a lot this kind of question what are the vector spaces that have the graded vector  spaces whose graded dimension these integer coefficients are encoded.  So for from a viewpoint of low dimensional topology this is a starting point for asking  more interesting question about vector spaces homology but first you have to have a Q-series  with integer coefficients.  And again I actually don't know how to associate Q-series to a three manifold or other objects  in low dimensional topology so I would be extremely interested to hear.  So one standard variant of three manifolds which is very powerful is Wittler-Russchitzelian  porive variant it's computed by Chern-Simons path integral exactly as Marcus explained  in his morning talk.  And it's a path integral over gauge connection A whose exponent is K times Chern-Simons action  as is the Chern-Simons functional that Marcus wrote on the blackboard and K is the level  integer that has to be integer because the integral needs to be well defined.  So in this story as well as in many other quantum field theories you can start Feynman  diagram expansion which is basically saddle point expansion of the integral this is something  that physicists do and love and that's precisely a good starting point for resurgence because  you obtain perturbative expansion or a saddle point approximation to the integral.  You have vastly growing number of diagrams in fact this is stolen from paper of Marcus  where this table and examples of diagrams show you the number with different number of loops  r is the number of loops maybe minus one and d is the space of such diagrams.  So like I told you in the beginning both Marcus and Maxim influenced a lot of what's going  on here so again I want to thank them for sharing that wisdom and hopefully not blaming  me for stealing the pictures from the papers.  So like Marcus said we can start doing perturbative expansion to this path integral and try to  approach it with resurgence theory.  So in this context we'll use two versions of resurgence theory one is roughly 200 years  old goes to Picard-Levschitz talks and so on and unfortunately it will not be enough  to treat some of the questions that arise in trans-seimant theory and its analytic continuation  so we'll have to resort to resurgence of Picard-Levschitz theory which no pun intended is called resurgence.  So that's only 20 plus something years old and again there are many key players here  in France so this will be version 2.0 of Picard-Levschitz theory that will be able to  track certain technical difficulties that we're going to see in this analysis.  If you're naively applying Picard-Levschitz theory to Schrodinger and Simons path integral  you expect roughly the following you expect that if you complexify your variables or fields  in this case you naturally will get expansion where you have subtle points that are various  flat connections and again if you complexify this will be complex flat connections on a  three manifold M3 and for each one of them you have this kind of trans-serious expansion  where you have exponential factor that I made very explicit is basically trans-seimant functional  of a given flat connection times perturbative series expansion in 1 over k so for purposes  of this talk I use several variables so q is exponential of h bar and this is same as  2 pi i divided by level k so the small parameter is roughly 1 over k.  So that perturbative...  So the L3 is real dimension...  Yes this is three dimensional manifold critical points or subtle points of this action functional  are precisely flat connections and M is the set of flat connections.  But do you say complex how do you complexify?  Because in resurgent analysis as well as in per-carolabschist theory natural thing is  to complexify the original variables.  So over the three manifold you have some...  Right. ...connections but connections are real?  Well originally we had g bundle over three manifold M3 and then naturally we passed to  gc bundle basically meaning that if you had connection A you now which had some reality  condition for example it was Hermitian, anti-Hermitian now you make it complex valued and declare  that all complex values of its components are allowed so just formally complexify.  Because doing this may help you find subtle points which are not necessarily on a real  axis and they still may have important contributions.  So you formally complexify everything you had before and then decompose...  Right so I should probably not say.  So there are two versions or two ways to look at such integrals using subtle point approximations.  One is where given each critical point which I denote by alpha here you associate with  all trajectories of steepest descent that Marcus mentioned earlier and they usually sweep  out some surface called left shift symbol so that's a picture of a left shift symbol  and then you can project this left shift symbol onto what will be called Borel plane.  This is basically projection by your action functional so S of A projects you to complex  plane so values of S is precisely what I'm going to call Borel plane later.  So the thing is so then what you get is this naively you would expect that your with-narration  to arrive in Borel in this case has expansion of this form where this is perturbative series  around each subtle point some over subtle points with some coefficients and alpha which  are related to what Marcus called C and determine the choice of the contour.  The problem is and this is one interesting thing where picor left shifts is not going  to be enough is that in terms of Simon's theory each flat connection has infinitely many lifts  and that's a delicate thing.  So the space of all gauge connections, modular gauge transformations is not simply connected.  You may go on a closed loop kind of like this and each alpha has infinitely many copies.  That's related to the fact that in four dimensions we have instanton number and that's also related  to the fact that transimons of alpha is the same as transimons of alpha plus any integer  namely the transimons function itself is only defined mod integers.  So what happens then is that instead of one subtle point you have infinitely many towers  in infinite towers of subtle points.  So that's already kind of tricky and that's why I introduced here this notation bold alpha  which denotes the lift of your subtle point into this tower.  This is interesting thing and I think I should have interesting applications to other problems  of trigonometric or hyperbolic nature.  So if you have two different equations, if you have trigonometric or hyperbolic integrable  system you'll have similar phenomenon.  You'll have sometimes this infinitely many lifts of a single subtle point.  If your equations are written in trigonometric form.  So that's cross that means that now we are working on the universal cover.  So instead of space of all gauge connections I have to introduce the universal cover and  Z tells me on which sheet of the universal cover I am.  Because you have second to next alpha theta on the first formula and alpha zero on the  second on the theta zero.  Right.  So theta will appear in a second and this bold phase alpha is again the lift.  So this basically says that if you had, if you thought that your manifold had say three  flat connections now what I'm telling you is that it has to have infinitely many copies  but ends should be such contributions of all of them, of all this towers should be such  that each individual and alpha and bold phase alpha sum up into what we used to call n alpha.  So n alpha is contribution of this connection alpha and bold phase alpha means it's integral  lift.  So that's first delicate point that you encounter again on gauge theory trigonometric hyperbolic  integrable systems things like that.  Q difference equation.  But your gauge group is here, is here to see and you're descending down to SU2.  Here I was saying that, okay sorry that's a good question indeed.  So here this last part of the equation says that originally this written Rache-Sichin  drive invariant is defined by SU2 gauge theory.  So our contour is such that all real points should contribute with coefficient one.  At least that's what we, well that's basically part of the choice of the constant C that  Marcus was telling us in the first part of the talk.  So it's additional data, it basically tells you how to set up your problem.  And the problem is set up in such way that you want the coefficients to be one if alpha  is SU2 flat connection and zero if it's not.  So that's the definition if you wish.  It's part of the definition of how we want to define trans-heimance path integral.  We want to define it by integrating over real slice in this complexified space of fields.  So that's basically the second.  It is not really well defined from this point up here.  Exactly, and yeah, that's what I believe I say on the next slide.  Indeed it's not well defined and this left-shift symbols exhibit the car left-shift monodrome  is.  So if you start there in parameters such as H bar or Q or other parameters you will have  stocks phenomena and stocks phenomena are basically saying that if you go from one configuration  to the other where I just draw for you projection on the Burrell plane what happens is that  a given contour gamma receives contributions from other contours and the intersection  numbers will determine for you what's the coefficient.  So that's another interesting phenomenon I want to emphasize.  In a finite dimensional problem the stocks coefficients or at least in Picard left-shift  theory these coefficients are integers because they're intersection numbers of some cycles  left-shift symbol and left-shift anti-symbol.  So what happens?  There may be zero cycles and you can you are integrating over the zero.  For the cycles which gives you zero.  So you must specify the cycles.  In usual Picard left-shift theory these cycles are mid-dimensional in the space of complex  fields.  So their intersection number is again if you're in a nice situation it's just a number.  It's an integer because it counts the intersection points of two mid-dimensional cycles.  So it's just an integer.  But I want to emphasize that here because of the first issue that we had infinitely many  settled points what's going to happen is that effectively this number will become fractional  and that's a pretty cool phenomenon because we'll have some of integers some of say plus  minus ones and they will be they'll need to be regularized resummed and as a result you  get some interesting fractions.  So this is another tricky point which usual Picard left-shift theory cannot handle but  resurgence does extremely well if it's very clever.  So that's I think where I'm heading next.  So you want to treat this kind of problem using resurgent analysis which projects everything  to the Borel plan.  You don't have to think about this infinitely many copies in the space of fields it will  do it for you by itself even if you are very naive and but if you care about what happens  in the space of all the variables you can do that too but it will take care of it.  And very naively what you do is the following you take your perturbative series such as  this H bar expansion you divide by r factorial which are this power loop coefficients exactly  as Marcus explained in his talk and produce Borel sum BZ of psi and then just like a number  theory you ask about analytic structure of this function.  You don't just do the summation you ask does this have analytic continuation to complex  plane and if so what are the singularities what are the residues very much like what  we did for L functions.  Now let's see so this picture is illustration of Borel resumption where you only sum not  up to infinity but just take first couple of terms and this is what you can easily do  with Mathematica if you're like me and don't want to spend too much time on either thinking  about abstract argument or even in Mathematica you don't have to have high power Marcus is  extremely good at this I know only how to do say five terms or three terms it's quite  amazing that this actually is very nicely behaved and pretty fast convergent at least  in good examples so this is done with very poor accuracy but you already can start seeing  that something is probably happening on imaginary axis that you may have some singularities  this is supposed to be profile plot for for this Borel sum.  Indeed what happens you have poles on the imaginary axis and again I want to emphasize  with the slide that what you have to care about to keep in mind is what are the singularities  on the Borel plane and what are the residues that basically at least for me tells what  resurgence is supposed to produce and encodes the most essential data about the problem  so it's like a snapshot like a photograph of Borel plane that that that says a lot.  So in this case again coming back to our example you perform lateral Borel summation and because  everything was on a all the singularities on the imaginary axis you can deform contour  and basically some of their residues but the only thing that's going to happen is you're  summing over infinitely many sedals so in the Carl Laphs theory it's very hard to develop  the homology theory or any sort of homological machinery which deals with this infinite number  of intersections and so on but in the Borel plane that's very easy we just have infinite  sum of residues and that's not a big deal and it produces for you some final answer  in the form again that Marcus explained and here I think this quote by Hallemard is very  appropriate especially in this occasion and right so the two key ingredients are the coefficients  and alpha beta which tell us how corrections to saddle point alpha will come from other  saddle points these are called trans serious coefficients and then position of singularities  on the Borel plane determines for you the exponentials this is precisely what Marcus  explained in his introductory talk.  So since he already spent quite a bit of time on exponentials I'll emphasize for you the  structure of this trans serious coefficients and alpha beta as I already mentioned they  come at least in this kind of problem or in any other trigonometric or hyperbolic problem  from infinite sums and you get sums of plus minus ones which are basically the intersection  numbers only thing you have to sum infinitely many times plus minus ones you can do it using  zeta function regularization you get this kind of expression which essentially then  boils down to some fractions okay and that's a very interesting phenomenon which in this  context leads to a miracle that I'm going to summarize for you in the form of a theorem  so this theorem comes from paper of myself Marcus and Pavel and it says something interesting  so if you have in the turn time is there you have different flat connections which have  different stabilizers so part of the gauge group may still fix your flat connection and  if that happens the flat connection is called reducible so it turns out if you think about  it from the viewpoint of the Corleaves theory or resurgence theory that reducible flat connections  are such that they can receive contribution from non-nibelian ones as trans series but  not the other way around so there is directionality so irreducible and reducible are not the same  at all and again part of this has to do with the fact that they have stabilizers so that's  an interesting theorem and the consequence of this theorem is something cool it says  that you can decompose your Witten-Rischy-Tiching-Turivon variant into some of blocks call them basic  classes for instance by analogy with Donaldson theory how it decomposes into cyberquidon  variants basic blocks or basic classes labeled just by Ibelian flat connections and it turns  out that these guys do have nice analytic continuation inside the unit disk even though  original fellow doesn't again I have no idea how to analytically continue Witten-Rischy-Tiching-Turivon  variant from roots of unity.  Ibelian means reducible.  Yes sorry I'm talking about SU2 group so for me reducible Ibelian are the same but in  general you are absolutely right reducible so Witten-Rischy-Tiching-Turivon variant  is only defined at roots of unity and a big question in this field is how to give a Q-series  again I don't know the answer to this question but what I do know is that to a given three  manifold you can associate not just one but collection of Q-series expansions labeled  by Ibelian or reducible connections each one is well defined in the unit disk and they  do have exactly this property that if you approach roots of unity from inside the unit  disk you get the corresponding value.  All the complex elements are away right these are all unitary connections.  These are unitary connections?  No no no this is complex.  This is complex with a point that if Q approaches root of unity they have to reproduce Witten-Rischy-Tiching-Turivon  which is the real slice.  I meant the connections are unitary connections or complex in this discussion?  In principle if you are inside the unit disk if Q is complex any Q with less than one norm  then all flat connections real and complex contribute to this expression so all of that.  But from the point of view of partition function we are calculating partition function of  chance simmers on M3 of unitary chance simmers or we are trying to calculate the sum.  But then it decomposes into sum of all possible contributions.  The function is of unitary connection but it is rewritten in terms of something which  is a billionized and complex.  And it is a billionized in a very funny way so again it is a billionized in a way where  you take so maybe that is another question I wanted to raise where you have.  I can't think about Z with a subscript a M3 as some kind of a billion partition function.  You can but I don't know how to do it in terms of simmers theory.  Using some string theory language you can.  So I feel that this may be getting a little abstract let me see a show of hands I am going  to finish in five minutes don't worry.  How many of you are happy with Picor Lefcher's theory just to make sure that I am not losing  everybody?  Just Picor Lefcher's theory.  How many of you are happy with Picor Lefcher's theory with infinitely many Picor Lefcher's  monodrome is happening at the same time?  Okay not so bad.  So for the rest of you that is the luxury of the physics talk.  I have a demonstration so I just want to explain what is going on here.  So we have a theory where we have infinitely many critical points, subtle points and let  me represent each subtle point.  So by bin.  So this is going to be reducible flat connection.  This is going to be irreducible flat connection and that's another reducible guy.  So to make it easier I will denote irreducible ones non-abilian flat connections by red crystal  balls.  There are infinitely many copies of each flat connection each subtle point because of this  lift to universal cover.  So I represent them by infinitely many stones, red stones sitting here.  So that's going to be reducible.  So middle one will be irreducible.  You have infinitely many copies because of this infinite cover issue or hyperbolic nature  of the problem of a subtle point.  And this is going to be another non-abilian flat connection.  So reds are abelian, white is non-abilian.  So what happens in this process of Borel resumption or resurgence is that this guy gets  distributed among the abelian ones.  So the contribution is still there just like it was in the Witten-Risch-Tichen-Turibund  brand.  Witten-Risch-Tichen-Turibund brand sums all of them with a coefficient one.  But what happens here is that first of all you have infinitely many copies.  That's the first phenomenon.  And second is that because of the Picorolevschitz monodromes that a plus minus one summed infinitely  many times, this guy, non-abilian fellow, gets distributed with coefficients say one  third and two thirds.  So what I should do to produce this object, ZA, on the right hand side, I should redistribute.  So these are now ZAs.  And they do have everything we had before where total now is divided just into reducible  fellows.  So hopefully this will clarify for you what Picorolevschitz theory with infinitely many  leftist symbols means.  And maybe as a question, and I think I should probably conclude here, I want to ask for  other phenomena maybe in number theory where the following happens.  You have some entity which is total, and for us this was total Witten-Risch-Tichen-Turibund  brand.  And in my example it was contribution of reducible, irreducible plus another reducible.  And the middle guy gets redistributed into these two with some coefficients say one third,  two thirds such that the total sum is still there.  And I want to ask if you know of any other phenomena, for example in number theory or  elsewhere, where some objects of slightly different nature, there should be some analog  of reducible, irreducible, maybe a billion, non-a billion, get redistributed in very similar  way.  For example, I could find L packets in the Langlands correspondence being somewhat analogous  where again you have certain representations and you package them together.  You group them in classes with certain coefficients which are kind of analogous to this trans-series  coefficients.  But I would love to hear more examples.  So finally, I don't want to take more of your time.  I want to point out that once you start with this perturbative series expansion in 1 over  K in trans-simon theory, you do this clever Borel-Ris summation.  Resurgence is very smart.  It takes care of all this pole summations and so on.  You produce this object ZA and these guys now have, first of all, by definition because  we were summing over steepest descent contours, they're well-defined inside the unit disk.  So this is actually Q-series.  And second of all, they turn out to have integer coefficients.  So some magic happens where for any three manifold in this trans-simon theory, not only  you continue your written Rishi-Tikhin-Turban variant away from the roots of unity inside  the unit disk, for each of these basic blocks, ZA, they start having integer coefficients.  As functions of Q.  So that's, to me, magic.  I have no explanation of this from a mathematical point of view.  Physics suggests why it should be the case.  But that's at least the result.  So it turns out that these are precisely mock modular forms that we saw on the previous  slide, at least in the context of Cyphert manifolds.  Probably not in general, but at least there.  Okay, I think my time is up, so I should stop.  Thank you for this next talk.  Other questions?  ZA, do not have integer coefficients.  So ZA are basically, yeah, so there is a question of notation in our paper.  There are ZAs and ZA hats, and there are simple linear combinations of each other.  So I promised you two theorems, and I managed in my time to get only to the first.  One theorem would tell you which linear combinations have integrality.  But that's, I would say, a slightly simpler issue.  Yeah.  Yeah.  Regarding the categorification of Q series, the first question.  So as far as I recall, there is a paper with specter.  He actually used this written index and derived the function as some measure of some physical  system, something.  So I think this kind of configuration exists in physics literature.  Then I think Allen Combs and Barnall, number theorist.  So he also has similar things, but derived from the specter's paper.  The specter of two papers, one is half-leap broken on half-leap broken supersymmetry.  And what is the Q series network?  I think he constructed something out of that, since I don't recall.  For some reason, information is just very common.  Well, I would love to have the reference.  And again, for some isolated things, of course, this is easier.  My question is how general would that be?  So here, for example, this is for class of mock modular forms.  I would love to have it for some big classes, not just one single example.  But anyway, I'll take over.  But this is for the zeta function.  It's somehow related to some green Hilbert-Fuller construction.  So, uh-huh.  OK, sounds good.  Thank you.  Very, very useful.  So, explain why the theorem one is true.  OK, so theorem one is true for the following reason.  And again, that should be also a good lesson for resurgence.  So what happens is you have subtle points, one, and irreducible guys are non-degenerate  subtle points.  Subdual ones are very degenerate because you have further action by stabilizer of the group.  So they're actually steki.  So what we're dealing with, we're dealing with Picard-Levschitz monodrome is, or crossing  types of singularities were.  A1, so to say.  Huh?  So you're saying it's not A1.  It's not A1, no, not at all.  Right, so from a singularity theory point of view, and in terms of assignments, this is  unavoidable because trivial flat connection A equals zero is always a flat connection.  And it's highly reducible.  It's all of the group G stabilizes that guy.  So in trans-simon theory, you cannot have a situation where all your critical points  are nice, more, so even more is bought.  You're in a situation where there is this big degeneracy.  And what's happening is basically the denominator, the quotient by additional stabilizer shifts  the Mohr's index in such a way that Levschitz symbols, if you think about it from Picard-Levschitz  monodrome point of view, don't even see each other.  So they're in a different grading or different dimension.  So from Mohr's theory point of view, they're in their own position, not mid-dimensional  anymore.  So that's the difference between reducible and reducible.  So as a smoking gun for a surgeon's or quantum field theory or anything else that you may  be interested in, if your singularities are not just poles or not just simple singularities  in Picard-Levschitz theory, there is some degeneracy, you should expect phenomena like  this where there will be directionality.  So in this case, again, reducible can pick up irreducible as pieces but not the other  way around.  It's all about singularities.  Is it independent of M3, right?  Yeah, yeah, yeah.  It's independent of M3 and in fact it's independent of this infinitely many copies.  It's just about behavior of Picard-Levschitz thingables for degenerate subtle points.  So and unfortunately or unfortunately, they're completely unavoidable in trans-heimance theory  so we had to deal with that.  You have one more question?  So at the beginning of the talk, you asked question one about some commercial origin  of as a stensiles.  You know there is a stensile symbol that gives you as a stensiles of the form that you wrote.  The lattice sum typically comes with a character in the numerator and it's associated with  cycles.  So for example, you can get a version of the E4 series with some character from three  cycles on the Kligow varieties.  It's called Beninson construction and from that you can lift up by competing some asian  or something that is induced by the homology or the homology of the main.  I see.  So there is some HMM, some doubly-graded homology groups that give you as a stensiles here?  So there is something that is attached to the cycles that is called the as a stensile  symbol that leads to some as a stensile with a homomorphic one with characters and they  are just put in place by Beninson sometime ago.  Well excellent, I would love to learn more about it.  Okay, so I hope that we stop now and we do in ten minutes.  Thank you.
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