 from Mainz and by the way, we're running a workshop in Mainz next week, so oh wait, you can watch. The deal is. Yeah, so it's not intended here. Oh, okay. Okay, okay. Well, it's a very exclusive. Forget about it. Forget what I said, forget what I said. You can, you can, you can, you can crash. These are not, these are not my rules. Yeah, there's an institute, the MITP has all sorts of rules. So the title is 3D BPS indices, Q difference equations and non-perturbatory corrections. Okay, thank you very much. Thank you very much for the opportunity to speak up this nice workshop. By looking at the titles I noticed that maybe I'm working a bit on the exotic topic here. I don't think that there are many talks on 3D. 3D gauge theory, so I decided to try to give a bit of a general overview talk. So this is a topic that I've been working in particular with my collaborator Peter Meyer for a couple of years now. And what I'm going to talk about today is mainly based on a paper which also appeared I think two years ago and some work which hopefully is going to appear. I'm not making it any longer but it should be appearing in the next couple of weeks. Okay, so, so what's the setup for the physicist, it's an n equals three dimensional n equals two gauge theory. And if you have such a gauge theory, then we can, we can have big branches. And for the mathematicians, what this is, there's a manifold involved which we call it target space. So it's a killer target space. It's a base 24x, which in this in the setup is realized as a synthetic portion or if you think more frankly about it, it's some, some sort of a GIT portion. That's a GIT portion that we've also seen in the previous talk. And a gauge theory comes as a, as a compact leak group, a gauge group G, and this GIT quotient arises from a vector space and the vector space is generated by the the Kyle fields. So the matter spectrum in this theory. And it's divided by the complexification of this with the reductive group that we are dividing by the complexification of the gauge group. So if you have such a setup, and this is mainly the quantity I'm going to talk about today, you can calculate a 3D index, or sometimes it's called a 3D path index. And so this is an index of boundary DPS operators. So let's put this theory on a three dimensional manifold, and this three dimensional manifold has a particular structure. So topologically, it's an S1 kind of this too, but it has a bit more structure to it so it comes as a metric in the physical So it's really a disk fibered over the S1, but metric wise the disk fibered over the S1 and when you go around S1, and rotating the disk and this is parameterized by this parameter cube. Okay. So if you have, if you want to count these things, then there's a certain index, and this index is roughly speaking the following, you have to take a trace. And since this is a, this is a space with a with a boundary. And of course, as a boundary, you have to, you have to take a trace where you impose on your field and operate a certain boundary conditions, and the kind of trace you can take. So it's an index. So your way. So for example, if you have a tracing, you have recreditation with Poissonic and fermion modes, you write this with a fermion number, and then they are some fugacity. So one is q. So q goes together with the Lawrence generator that you have in the theory of this disk together with what is called an asymmetry generator and possibly additional symmetries you have in the theory called labor symmetries. You have to take the kind of index, so you can compute so after is the generator of the flavor symmetry. So it's as you can think of this as a symmetry of this, the quotient. J is the spin generator on this disk and are there is an additional symmetry in the super symmetric series. So it's a generator of the corresponding to one our symmetry. And certainly not an index that that that we have looked for the first time. So this has been introduced for the first time by beam. In 2012, then the group of and put off in papers in 2013, various contexts have studied this index. And what we are mainly using is the formulation. What am I in more. I got your job from 2017. They are many other direction that's on top. That's kept tea. That's kept. It's not the same question. Not as ah, sorry, yes. You can look it up. So they are, they're different, they are different people and yeah it's okay. Okay, good. So for the purpose of this talk, I will actually use a particular simple setup with a simple setup you can think of concerning the gauge group. So if I just take the gauge group. If you want to be one. Then the kind of g it questions we are studying a central very basic examples of growing varieties. And, and moreover, like I said we have to impose boundary conditions on the various fields that we have in the theory. The gauge theory comes with a gauge post on in a particular we also have to impose a boundary condition for the gauge post on and the one we choose or the one which is kind of the riches to think about is the technically boundary conditions conditions for the gauge post on. So if you do this, then you can study. This is for this particular setup, the index takes a very specific form. It now depends on free velocities or it could depend on more but these are the gas that we interested in. So, Q is this parameter which we have universally in this index that will keep the queue is is a global symmetry, which is called in three dimensions. You want topological so that's the dual symmetry to that to that gauge group. But this is dual. G, which in our case is the one. This S and S has to do with this choice of boundary conditions so if you impose directly boundary conditions. It means that we don't have a gauge post on at the boundary in particular because we have set it to zero with this directly boundary conditions. What we have to notice is that since we have this this this gauge symmetry G, which in our case is you won, you still have a remnant of this at the boundary as a global symmetry. And that global symmetry at the boundary is usually in order by you, you want partial, and this is nothing else but G. Okay, so we have a quantity, which depends on these on these three variables. Then this big Q and this S. Okay, so this has a character characteristic expansion. And then there are some quotient functions like and that's you. And let me say a few words about about what you get here. So this is, this is a sum of a boundary. And this is one of the multiple sectors that you have in this theory. This, this whole thing here is a Q. The lateral series. So it is formal. It's a formal formal series in Q and you inverse. And this, this, this part here is just something that comes from the matter of fields. And one thing that I haven't talked about, they are in three dimensions, they are particular terms you can add to these to these theories topological terms, but they show up in this index. They have their so-called chance I mean search for their specific choices. So, how does this look like in practice. Let me just give you an example, or let me just be schematic here so how do the terms look like. So that gives you a factor of Q to the one half a times m squared. So we have a B M where a and B are choices of transignment levels. So a is is you want for us, you want to and Simon's level and be the one times you want our terms items level. And they're suitably quantized. They're both active trends. We'll put the fact that time level, and also how does. How does the matter index look like. Very concrete formulas, matter index for a Kyle. So again, we have two kinds of situations. Either we can impose Neumann boundary conditions say to the scale of field in that multiple and then the boundary conditions of the remaining fields of these multiple are governed by symmetry. And then what you get is something as alpha. Q to the alpha m, Q to the r half, in terms of these Ohama symbols. So this alpha here is the U1, the U1 charge. So the representation with respect to the gauge group, S is this forget that we had and Q is this little Q. And if you have directly boundary conditions. And then you get a similar expression, just, sorry, this is to the minus one, then you get a similar expression, but you get it in the, in the numerator. And one minus r half. So what are these Ohama symbols. Ohama symbols. So for the magnitude of this variable q which we take to be. Here it's still a formal variable, but we can think of this as, as, as, as, as a complex variable, which, which we choose to have the magnitude smaller than one. Another choice would be bigger than one that's actually a girl and at the end of the day, but that's our choice. But if you do this, then the Q half Ohama symbol, just nothing else but this infinite product. And then what do you so depending what, what spectrum you have and what choices of terms I mean terms you have, you can immediately calculate this index, which is, which is taking this formula and writing it as a product of all these contributions that you get from all these matters. Yes. These, these are examples are these. So in this theory where we have the gauge theory it's the one. So if you have, this is the contribution of a single carol field. If it has no one boundary condition. It's, it's one of our popular factor of that work. If you have directly boundary conditions, you have, you have this factor in the numerator, and I should have mentioned. So our is the you want our charge alpha is the gauge charge. Yes. Yes, so here appears the index and so you have to sum over. This is a natural formal display lateral series. It's just this infinite some for the integers. I mean what you could do okay so I mean certainly you can talk about more general gauge groups and more general GIT quotients and then you could also calculate that index I will give you a geometric interpretation of that index in a second. Yeah, good. Yeah, so, so this why stands. So one if you want one global. So this stands for labor symmetry and the global symmetry. So one example in this case so you should think of this why in this particular example just of the Q and B s. In general, you're right. I mean there could be more labor symmetries and particularly for your story varieties. You can look at the equity variant setup, and then you would have, you would have for every C star symmetry of your story variety you would have an additional why if you want to keep track of it. Yes. Now a is just in this case. It's just in. It's just integer half integer. So, let's talk about properties 3D index point. So, one, one result that we have, again now in this work in process we make more precise button, it also has been already worked out in collaboration with my and we need that Alexander Pablo is that if you have this X, you have this killer target X that, like I said, if you just think about it, I directly think about it. There is an identification of this you want partial. You can make that you identify as this key for the minus one, which is nothing else but the churn character of age, where H is the hyperplane bundle of X. So in this case where we have a single you want this. We also have a single as a gauge group we have a single you want partial and there is also a single H in this setup, but it certainly has an immediate generalization to more general very great. Or even more general GIT quotients, then we can look at this index. Now given in terms of where we set S to this churn character there or to this inverse churn character q, and then this is proportional up to some numerical factors which I'm not putting a track of the purpose of this talk. You can expand this out into a quantity I of X, which now depends on this. It's H or what I mean by this the churn class of this hyperplane one. Plus, and then, since, since, since this is, well, if you think of this as this divisor, then this automatically truncates as the dimension the complex dimension. I mean the highest form you can have this is complex dimension, this manifold X, so all the terms that are higher in degree of age, they vanish upon this identification to do this expansion. And this I acts of age comma. You, this has a very geometric meaning. This is sometimes called a period. It's called the key theoretic. Given I function of this, of this target space of this GT quotient X. So what is this I function. This I function. I mean we have kind of heard it was in the previous talk the homological version of this, this I function is a generating function of certain invariance you can calculate on these target acts and what are these. These, this is a generating function. This is a generating function of polymorphic Euler characteristics on the modular space of stable maps of genius zero. And I write an example that say this one map point of X, which is the following form. So Lee has constructed a student of him tell has constructed a structure sheaf. I should say that the professor in the field has established a virtual structure sheaf on this modular space of stable maps and this allows you to actually calculate Euler characteristics, which are the following form. So if you have on this modular space of table maps one mark points, you can take for instance the evaluation map. And it's like this, ours, he was on power, take this bundle is the line bundle key, some power K. And divide by possibly in this is where this Q shows up, where we have this technological bundle on this particular space of stable maps, but this is the code tension line that we get from these mark points. These are Euler characteristics, which are encoded this generating function. So, these numbers are integers. And just as a little side remark, you recover from. Yes. This is the modular space of stable maps from Tina zero is one mark point into X, the virtual dimension of that guy depends on. Well, I don't have the formula or the top of my head it's a usual formula that you know and lock. I mean that's, I don't know. That is right I mean this formula depends on the first, I mean the virtual dimension of this formula depends on the first chunk of the dimension of the first turn class and of the genius and this generating function that we are discussing other genius zero. Okay, just as a little side remark to what we've, what we've seen in the previous talk. So if you send q to one, then you recover in this limit. You recover quantum homology. And as such, what you also recover. You, you, you, you recover rational. The rational numbers rational genius zero from a fit in invariance. And, well, a particular instance that we physicists love to study is if you have a color be our threefold, then we know that there's a re resumption of these from a fit in invariance to go for come a buffer invariance which are integers. And this resumption is kind of based of also relating these grimo fit in invariance from a targets base perspective to to one dimension higher by adding an additional as one. And this, this Euler characteristic, yet another resumption of the grimo fit in invariance, which are integral, but they are not specific to any club house anything that works in general. And this is a kind of a form of physics perspective a very similar idea, because there is no reason for the grimo fit in invariance to be integral because they don't stem from an index. However, if you, if you calculate, if you calculate these, these generating function of this counting function of the operators, then it's very natural that they are integral numbers. And so what we're saying is that that these integers from that perspective. They kind of a world she world, world she world she theory left from two dimensions to three dimensions by also including additional one, which make them integral. No, no, no, you have to be careful so the the Euler characteristic is always integer. We are not doing intersection theory. So, actually, you're raising a very good question since this is a stack so you have possibly some overfold action on this. It's a bit tricky and this makes it actually also complicated to calculate these numbers to define these numbers properly. And there is what is called an inertia manifold that which has several components that the attribute to the stack. And then you calculate contributions from each component in this inertia manifold and that is always guaranteed. Yes, so you're right. So yes, indeed. Yes, yeah. Thank you. Okay, so these are actually rational. I mean, integral coefficients with rational functions in Q but just to be in the safe time. Okay. Yes. Yeah, so this is in general so you're when you take the Q to one limit, you always recover the quantum pomology invariance the chromo fit and invariance. I'm just saying that in the case that this is calabi I mean you can do this also for fun or in principle if you're powerful enough to the results that we are previously there should also be a limit. So if you look at the picture constants of these rings that we talked about in the previous talk, they should also rise in that they should also rise in that. Is there a relation to higher genus so what is known. Okay, so this is an interesting question to higher genus for color via three volts. Okay, we can also discuss that afterwards a bit, but for color via three volts in principle these, these things I have been defined. It's difficult to compute them. Let me put it this way. I mean, so far. How you calculated. Okay, so you one way is you calculated by localization or the other way that I've been sketching here. You just calculated from this index. So we know, so you look at this spectrum, you can write down this. So either you do a direct counting of these BPS operators or you get it from localization. Okay, but for you one series are for. Well, I mean more general for general group G. I mean these these formulas are known so in principle you can as a starting point you can write down this index. When we set q to minus one. I think sending q to roots of unity is an interesting question and I'm afraid I don't have a good answer for this, and I've been asked this many talks, but I don't know what, what, how to interpret geometrically Okay, today I don't want to really talk too much about these, these invariance, but I want to look at it at more the analytic properties of these of these indices. Sometimes sporadically, and we also don't understand the system ethics of this so there's an interesting connection to this this block group, and these CFTs that show up in this nouns list of CFTs. And we get these modular functions that we can construct from the local considerations sporadic or specific choices of age groups and matter of the other in terms of levels, but we don't understand the systematic levels. Maybe you can also discuss what's going on. They are, I think they are relations to that. Yes, but I'm afraid I don't really have to. Okay. So, what's interesting to know is that these, these, these eyes one. So these I axis, they fulfill to what is called a Q difference equation equation. So in other words, so we have a Q difference operator L, which is a finite some. So this difference operator has some order and, and it has some coefficients. So Q, Q we, we think about having a particular value, but you can also think of this as a polynomial both and Q and a big Q and little Q. But if you set little Q to a particular complex value which is absolutely small value than one then these are polynomials in in Q. So if you have this, this difference operator, Sigma K, where Sigma acts on Q, or if you commute it has Q, it just multiplies it was a little cube. Okay. And now the statement is that. And I guess I want to know, I'm going to zero and I'm going to zero. So you can find such an operator that these, these, these I functions, I function on the lead. Okay, so this also has a physical interpretation. So this, this, this difference operator is actually award identity. Well, Wilson lines, lines in this maybe gauge theory. Okay, but let me, let me maybe focus a bit more on this difference equations. So actually, there are nice analytic properties of solutions. There are a few difference equations. And one thing is actually, and maybe as a reference. I mean, there's also many recent literature but many of the things I learned from some old work by the way from 2003. There's also by so long and collaborators and this was a part of the PhD thesis PhD thesis. This, this has been discussed. So one, one important aspect of these two different equations is, so if you have a meromorphic solution. Say in some open ring around zero. You smaller epsilon. Then what automatically automatically follows that you have a meromorphic you can extend this to a meromorphic uniquely extend this to a meromorphic solution over C star. And this is this peculiar feature. And this is one of a few difference equations. So, if you think about this what does this, this difference operator do. I mean it is, it essentially just replaces the argument of any function. By multiplying by little q, but since the magnitude of little q is unequal to zero. In our case it's smaller than zero by acting with the sigma or the inverse sigma, you can gradually always analytically extend using this difference operator. This to the next epsilon step. The only thing you're going to do is, since these polynomials in q, the worst that can happen is that you introduce by dividing by these coefficients, you introduce another poll, but that doesn't violate the property that's going to be a meromorphic function so. So if you have a meromorphic function in this in some small ring around zero, it automatically analytically extends to meromorphic solution. So the other thing similar what you use for differential equations you can also write down a first order matrix equation. First order matrix equation. Sigma F some connection matrix a which also depends on you but like I said q is think of a fixed constant for now. Where in the usual way, where we write, where we write this vector F, where we try to find the solution over it F is a solution then the vector after the solution this type of minus one to the F. Okay, so if you have now a set, the complete set of local solutions. Say X zero F one through and so if you have an end order difference equation and they're also in solutions at say zero smaller q small epsilon. And let's say we have another set of solutions. G one. So G and at zero smaller to inverse epsilon. Then we already know that these solutions by what we've said before, they all. So they should be meromorphic it should say morphic also meromorphic. We can automatically extend to solutions of whole of C star, and one consequence of this is that we must be able to rewrite this, this matrix of solutions by some matrix P, and this P is called a book of matrix. So this P has a specific property FB act with Sigma on P. Then it must be invariant because if it were not invariant, then you can easily convince yourself. If this is a solution and that is a solution of a consequence this P has to be invariant. So here, consequence of this. So we have now a matrix, translating these two solutions which is invariant and the Sigma. So a consequence of this is that that P is actually a matrix of elliptic functions. On the elliptic curve, it's how, which is C star divided by q to the C, q, q, and this is one reason why it's nice to choose q is more than one, you can identify so that the complex structure parameter of that elliptic curve is the function of the exponent of this of the cube. Yes. Yeah, so in small, so in small q. This is actually also an interesting thing which has was related to this previous question. So at least for these in the context of these given functions, it has also a very specific poll structure in small q, namely it has only polls at risk of unity. So, yeah, but yeah, that's, yeah. Okay. The infinite products I wrote down. Yes, so hold on. And that that that is true, but it has. So the poll structures you get. So okay so this depends a bit what you plug in for these gases but if you said all these accurate in parameters to zero at the end. The polls you get, and this is very specific to this case theoretic give and try functions. You only have polls and little q along risk of unity but yeah. Okay, so I'm actually this. So, so this is very nice to know that this, this matrix P is a matrix of elliptic functions but well I guess you're not powerful enough but for many examples, even though we can determine these local solutions. And because these local solutions are often given in terms of an infinite series. So we know that they are meromorphic functions over C star. We have a hard time finding analytic expressions. Actually, for these, for these specific for these book of matrices. I mean working out these elliptic function in general is in general not, not so straightforward. So some simple examples and one example I would like to present here because it's particular simple for the Connie fold. And I tell you in a second what geometry I mean by this, you can actually calculate that P explicitly and then there are a few other examples that we can do this. So, this consists of four Carl fields in this case. And again, the gauge group was you want, we said, so it has a single you one charge. So, we have Kyle fields plus one plus one, and we have Kyle fields minus one minus one. So for this simple geometry. We have actually this theory has two Hicks branches. And they both, they're related by geometrically they're related by a flop. And they look. The two GIT questions you can construct out of this, that the geometric are these are these two places, which are these results. And now we can go ahead and go through this program. And actually, so I mean write down the corresponding operator L solve the solutions so let me, let me just give you a flavor of this. So the solution at q equals zero. How does this look like. So, so there's going to be if you do this. So this is expansion in in age, if you do it here. So you have a zero and a two form class. So this is morally speaking, age to the zero, age to the one. We also would like to have some classes for these for these non compact direction and this goes back to I think questions, we actually have to go to a covariant setting now and calculate them a very formalogy classes. But more or less speaking, if you do that, what we can extract is also four form and six form class with compact support. And then to restically, to think of this as for bonding to something like it's two squared, it's three squared but really what we're calculating you know some equity variant pomology classes but let me not go into this discussion here. And then, if you if you do this whole thing at your end and at the end you can again send the equivalent parameters to zero if you want, then we get a period solution around zero. And this takes the following form. So one, in this case is a solution. Minus LQ is a solution and tell you in a second what this is LQ over two plus a huge generalization of lead to, and then, let me not write the third one, because it gets a bit more involved so it involves a generalization of the acute generalization of the three. So what are these things. Well, log Q is essentially the logarithmic derivative of a theta function of a theta function on this elliptic curve. And the important thing of taking this logarithmic derivative. The theta function is this is meromorphic, the way we've written it meromorphic in C star. And then if you take the logarithmic derivative we still get something more amorphic in C star, but the interesting property is if you if you act with this difference operator on LQ, then we get back LQ plus one. And that saves a bit like, like the monogamy of a logarithmic when we act with the Q difference operator. Okay, and like I said, this lead to Q is a certain generalization of lead to in the sense that when we send Q, again to one to the chronology limit, you get the ordinary lead to, and it takes the following form. The series one one infinity. Okay. And then there is also a choice of transiments the choice of transiments level enters here that I mentioned before. And then there is one. Okay, spread. And like I said there appears something a generalization of the three and the third line. And this has the characteristic features it has denominators which go with the cube. And again if you take the Q equal to one limit, you get lead three of that, that function. What do you mean this ordinary. So you mean the father of quantum dialogue. Now this will appear later on. So this is not just this is not just a pohama symbol. This relates to. So if you start differentiating this, you get similarly, similarly like you are from the ordinary. And you just start differentiating this you get, you get this L cube, then you start differentiating this then you get a pohama symbol. Okay, so now you can do the same story for the, for the other for the other solution as well. Let me not go into this but let me maybe. I'm running out of time. Let me just make an observation so if you, if you do this we can now work out. I mean, we work out this solution the solution at the other end we can work out the. And again what I should maybe stress, it's this function, which contains the the chronological that the case the erratic grimoire put in your answer the K invariant these are these are the characteristics that I've been talking about so this is the function which contains the information. And similarly you can extract it from the lab here as well. Okay. So, one observation is that the spirit of matrix of matrix. Cancels holds a solution holds in the solutions infinity. So, when we do this, for instance, for the poor form part. Namely for the part that I've written down here rather explicitly, you get the following formula that you can write this particular period period in the following fashion. So here you get an elliptic function, which is essentially given by the by a constant plus the wire stress p function. I've written in this multiplicative variables, times the period one, which we also have on the other side for you inverse equal to zero, plus, plus a minus one so minus one is also clearly an elliptic function a constant is an elliptic function. Where q tilde is essentially the same quantity that we had here just written in the inverse variable that w, that w is the inverse. So now what we see here is the following property. So we first of all we see that we have two elliptic functions that these are the ones which come from the book of matrix. And then the other thing we see this that this actually when you analyze these functions more carefully. This is holomorphic at q equals zero. This is holomorphic at w equals zero. And these, and, and these they are meromorphic meromorphic functions on C star. So what you see is that the poles of of this of this function, they get by these meromorphic functions, they get canceled because the poles that we have here if you go further out in q. They are not up here so this thing has different poles. And the, the, the, the job of, of this, this factor but with this essentially comes from a John Simon's term, but a particular of this factor is to cancel the poles that we have in one expansion with the poles of the, of the other expansion. Okay. And unfortunately I'm a bit running out of time, but let me just say, how we now interpret this or how we use this. I'm going to use a bit to Murat's, well maybe Murat's question. Yes. What the little. Now this, this important this does not have a logarithmic French cut. It just has an essential single. So the price you pay is that it has an essential singularity at q equals zero, but it behaves. So if you use this as a logarithm if you send little q one or, or as a matter of fact if you send little q to one. Then this is an interesting story in its own right this is described already in this thesis by so raw. If you, I mean, it's a bit. It depends how you precisely take this limit for these functions q goes to one, but then what you get is is actually a logarithm. Just maybe conclude by, by saying the following. So if you have now this equation. Um, so this is maybe part three. Non derivative corrections. We can now in this works in general, but for the conifolded bricks because particularly nicely and you can carry it out more. Yeah, essentially analytically. So if you introduce also as dual variables variables, which come from an S transformation maybe q tilde is if you take this elliptic curve and do an S transformation of the modulus of this elliptic curve. There's a natural associated q which we, let's call this q tilde which is q divided by one over q. But this is essentially e to the two pi i u divided by tau q is e to the two pi i u. And similarly, there's q, which is e to the two pi i divided by the minus one. And, and this q has the property if these act by construction if you act with this difference operator on this q, it actually commutes or it gives me to the two pi i u tilde sigma. So, you can also use functions in this q as as as coefficients in your series expansion and if you do that, then you, you can write down new solutions. Which have nice of the analytic properties. Let's say the, the, the solution of this form around zero. It now is a function of q and q tilde. It, this is now an ordinary logarithm of q, which we can attribute to a transience term. And that I can tell you afterwards more about this then we have this q that we had before. But then there's another piece which let's call this l plus, which collects the polls. Which collects the, the polls which enter in order to generate, I mean to communicate non polls here to pull there we collect all these polls bring polls to this side and the other to the other side. And then we get a new, we get a new solution. This is also a solution to this two difference equation, but it has nicer analytic behavior because it has no polls. But it still has, and this can, like I said, can be pretty attributed to transience terms it had, has branch cuts with respect to q tilde and w tilde which is the inverse of this q to the minus one. Okay, so let me just say what what this interpretation is so they are a couple of comments if you do this. It actually does diagonalize the book of matrix this is another way how you can describe it. It also has an interpretation so these, these variables also have an interpretation of physics interpretation. So this q was related to inserting. I mean adding with the q difference operator corresponds to inserting a Wilson line acting so there's a corresponding dual difference operator. It also corresponds to inserting a dual war text line, and then, in a certain dual symmetry. So what you're saying is that the non perturbed completion contains these, these, these dual war text lines as a background. And the other thing that there's a melon Barnes type of interpretation and if you describe a certain contour, which comes from a deepest this descent method. You also automatically collect these terms, unfortunately have time to talk about this so there's another way to derive this. Examples like just a single carl multiply which is the simplest, what you get is the, the Fadeff quantum dialogue and as for instance described in this paper by gaurib. And, and so maybe we actually hear more about this in your talk, because it turns out, if you look at the conifold the conifold this generating function they actually coincide for the conifold geometry. And it's not totally clear to us why this is the case with the topological string partition function, we have this little q corresponds to the genius parameter, but just for this case. And then it corresponds to the to the. I mean, so, so the completion that we got here is similar to the completion that more intense in his, his works as well so maybe let me stop. Well, we've had a lot of questions, but there's time for exactly one more before coffee. Sorry so you mean when we do so you come back to this counting problem. Yes. Okay, so, so yeah so this is one way how you can interpret it you can actually also. So this is kind of a land of Ginsburg kind of formulation in our first week, we had speculated a bit about this. What's the generalization of the holy water mirror. And they are there structure wise which things which look quite similar. So this is related to this melon Barnes type of story that I haven't really time to cover. So there's also similarly as as as in to the there are certain integral representations of these periods. And this is. So to say by some contour integral to rescue kind of prescription, another way how you can describe and this is maybe analog to the model, how you also can describe period. So if they arise really from this index, or from this localization perspective so you don't need to talk about these periods if you're just interested in this invariance but these, these generating functions, they will build this to different situation than you, then you can go from there and try to get the mirror. Okay, let's thank Hans once again. So, for 15, we have a talk by David's of arrow, it's on remote. So we have to be here on time.