 So, I'm going to talk about non-products for the political strings, and this is mostly started with this paper, with Alba Grasso, with a photo here, and my photo, just a few years to that, two years ago, so this led to many developments, and today what I want to do is to explain how this develops. So what I want to do today is to explain you how this work that we did gives an unperturbed definition of the political strings, and what I want also to explain, if time permits, is how this is related to resurgence, and this is working in progress with Ricardo Coso, Santa Maria, Ricardo Scam. Okay, so let me give you the sharp problem that we would like to solve. So as you probably know very well, string theory, and in particular the political string theory, is defined by a formal divergent expansion. So this means that if you take all the genus G-free energies, say, of the political string theory, this is an object that's featured just in Albrecht's talk, and you sum them over all genera, with appropriate powers of the string coupling constant, this thing is just a formal series, doesn't make sense. You just put numerical values for T and for G, you just get infinity. And the reason is that these FGs grow factorially with the genera, okay? And this is actually something which is very common in quantum theory, which is that perturbative theories typically are divergent, are convergent, are asymptotic, and divergent, and you don't know how to make sense of a total sum over genera in this case. So the question we want to ask is the following. Is there a well-defined and computable function of the moduli and the string coupling constant which leads to this expansion as an asymptotic expansion? And this is the problem of formulating string theory, non-pertorotic. So what you want to do is to actually interpret this formal series which doesn't make sense by itself as the asymptotic series of something well-defined. And it's very important that something is well-defined, because if you end up giving me something which is not well-defined, then it's not really worthy of thought, because the series was not well-defined to start with. So you have to give me something which is manifestly well-defined. And this is the problem of formulating the political string theory non-pertorotic. So the expression non-pertorotic of political string has been used a lot in recent years, has been abused a lot in recent years. So I want to actually explain what I mean by this by going back to basics. So let me explain what I mean by a non-pertorotic completion by going back to quantum mechanics. So imagine that you take a Hamiltonian of the quartic oscillator. So you have the standard harmonic oscillator, and you perturb it with a quartic term. So this Hamiltonian is no longer exactly solvable. And what you can try to do is to compute, for example, the energy of the ground state as a power series in this coupling constant g. So there is a well-known procedure, a formal procedure to do it, and you can push this computation quite far if you want. And what you're thinking this way is again a formal power series in g. So these are the first terms. This one half is the energy of the ground state of the harmonic oscillator, and then you have a formal power series in g. This formal power series is again divergent. It's factorial. The coefficient is diverged like n factorial. So again, this series is like the series of the string theory in the sense that it doesn't make sense. It's a factorial series. It's a formal series. You cannot put g equal to one number and get a number out of it. So how do you define the non-pertorative problem for this elementary example? Well, you want to have a non-pertorative completion of this pertorative series. You have to use a non-pertorative definition, which is provided by the spectral theory of studying your operators. This is what you need in this case. So what you have to do is to consider the Hamiltonian as an operator on the Hilbert space area to r, and you have to prove two theorems in order to give this non-pertorative definition. The first theorem states that this operator h inverse of h, the inverse of the Hamiltonian, is a compact operator. It's actually a trace-class operator for all positive values of g. So this is a theorem you can find, for example, in the book by Berezding and Subin. It's a proof. It takes actually a few pages. And then once you have this theorem proved, you know that this Hamiltonian has a discrete and well-defined spectrum. And the lowest value of this spectrum is the ground-state energy, which is then a well-defined function of g. So that's the first thing. So you see, in order to have an operator definition, you have to typically prove a theorem. It's not just enough to make some construction. You have to really, because you want to have something well-defined, so you better have some theorem . Now the second theorem tells you that this function that you define through spectral theory has an asymptotic expansion, and that this asymptotic expansion agrees with the expansion that you find in perturbation theory. So this is how you do well done. You have a well-defined non-pertorative compression of a asymptotic series. And of course, you want this quantity not to be a mystical object but to be something computable. So what you found in this way through this non-pertorative definition, this object that you define through spectral theory has to be computable. And actually it can be computed. For example, by using numerical deconalization, now I put g equal to 1 and I get a number. Something that I cannot do if you give me just asymptotic series. If I put g equal 1 in the asymptotic series, I get infinity. Here I get a number. So this is what I mean by a good non-pertorative definition. So this is the standard definition that you would get in quantum mechanics. Now, there are more complicated examples in the sense that you can have now problems with two coupling constants, not just with one coupling constant. And this is the typical case in ADS-CFT. Now in ADS-CFT, you say that you define non-pertoratively a strain theory through a gauge theory. So in the standard case, you have again a genus expansion, say for the free energy of a strain theory, of a super strain theory, this depends on some radius or length scale of the background of the theory. So for example in ADS, this is typically L over Ls to the fourth, where L is the ADS radius and Ls is the strain length. Then you have the standard genus expansion. And you say that this object, which is again divergent and ill-defined and so on, is the asymptotic expansion of a gauge theory free energy, which depends on N and Gs, and lambda is going to be related to N and Gs through the standard relationship that you find in ADS-CFT. So this lambda is going to be identified with the top coupling of the gauge theory. Now, see this is again an asymptotic expansion, and this defines the super strain perturbation series as the asymptotic series of a well-defined object. Now there is here a small subtlety, which is some sort of mismatch of the ranges of the variables involved. And the mismatch is due to the fact that N here is a positive integer. So you have a positive integer and a real number, while here you just have two real numbers, lambda and Gs. So it's clear that you cannot cover all the range of lambda and Gs by using a positive integer and Gs. So in a sense it seems like the nonperturbed definition, which is given by a gauge theory, covers less space than the standard super strain perturbation theory. So sometimes people say, well, but you know, when N is very large, it's almost like it was continuous, blah, blah, blah. Okay, that's nice, but we need sometimes more precise statements about what we mean here by a nonperturbed definition, because we would like actually to extend the right-hand side to a full function of N for any value of N. And this is something that I will comment on later. So just keep in mind when you do nonperturbed definitions through ADS-CFT, that one of the variables in the nonperturbed definition is intrinsically quantized, because it's the rank of the gauge group. So this leads to some subtleties that have to be others. Okay, so let me again give you after this discussion the definition of a nonperturbed definition. So a nonperturbed definition of a formal perturbative series is a well-defined function, which is computable of the relevant parameters, at least in some range. So at least in some range we want this function to be well-defined and computable, and this function has to lead as an asymptotic expansion to the original perturbative series and nothing else. You cannot have the perturbative series plus other perturbative series, because then you have more stuff than you used to have. Now, in the case of topological strings on toric alavigiaus, there have been some nonperturbed definitions in the literature, but I claim, and this is maybe a little bit polemical, that none of these definitions satisfy this criterion and applies to all toric alavigiaus simultaneously. Maybe I'm wrong, but if I'm wrong, I'm very happy to discuss it. So there are some definitions, for example, like large chain type definitions, alagopakumarbafa, which actually satisfy this criteria. They are actually very good in the sense of ADS-50, but they only apply to a discrete family of alavigiaus. You have Resolve-Connifol, AN-Fibration-Sover-P1, but it doesn't apply to all of them. And then you have other definitions in quotation marks, which actually do not satisfy this criterion. For example, definitions based on OSV, which express the result in terms of, at the end of the day, you can express the nonperturbed definition in terms of QD4 and MILS, do not satisfy this definition because the asymptotic expansion doesn't lead just to the original perturbed series. There are some extra stuff that you have to remove by hand. And then, you know, there are not really nonperturbed definitions in the sense that I'm using here in the standard sense. So, what I'm going to do in this talk is to provide you a definition, nonperturbed definition of topological strings on general toric alavigiaus models. So, let me review a little bit what is the main setting. So, the toric alavigiaus are essentially the simplest but non-trivial alavigiaus 3-4s that you can have, and they are all non-compact. Now, one technical way of giving you a toric alavigiaus is by giving you a Newton polygon. And in this case, if you are giving this Newton polygon, you can construct the mirror manifold to the calavigiaus by just computing the Newton polynomial of the polygon. And a very important aspect of toric alavigiaus is that the mirrors are just algebraic curves. After, you know, in general, as Albreg was explaining yesterday, you have a mirror of a calavigiaus 3-4, it's going to be another 3-4, but in local cases, it turns out that this 3-4 reduces essentially to algebraic curves. And this is very crucial because it reduces the complexity of the problem enormously. So, one very important example that I will use in this talk as an example to provide you with concrete details of the theory is local p2. So, this is a non-compact calavigiaus, a toric alavigiaus, which is the canonical bundle over p2. And the Newton polygon in this case is given by this triangle here. So, you have three extra external vertices and one internal vertices, and then you can compute the Newton polygon of this polygon by attributing to this vertex 1, 0, e to dx, attributing to this vertex 0, 1, e to the y, and attributing to this vertex minus 1, minus 1, e to the minus x, minus y. And then this vertex, which is 0, 0, gives you just a constant kappa, which is the modulus of this toric alavigiaus. So, this is the standard equation for the mirror of this, which is an algebraic curve. It's actually an algebraic curve of genus 1, but it has the subtlety that is expressing exponentiated variables. Okay. Now, once I give you a calavigiaus, toric or not, you can actually compute this topological stream free energies, a genus G, which usually encode what are called the grommet-witten invariance of these three fours. So, this is a formula that I think Albert wrote yesterday, and restricting the scalar class just to simplify my notation. So, the genus G free energy can be computed at large radius as a sum over all possible degrees of curves in this calavigiaus. The weight of such a curve is e to the minus dt, where t is the scalar parameter, and the coefficient is the grommet-witten invariant of genus G and degree d. Now, this is the standard definition in what we call the large radius frame. But, as in cyber-witten theory, a calavigiaus manifold can be defined in many different frames. And this is, as I said, as it happens in cyber-witten theory. These frames are all equivalent, perturbatively, and they are related by sympathetic transformations. Actually, you can relate the free energies, the genus G free energies in one frame to the other by using an integral transform, which was essentially based on this wave function idea. So, we did not have the wasspraming yesterday, but it was implemented in a very easy way in this paper by Nage-Boussard and Clem. So, these large radius frame of topolecal strings give you the grommet-witten invariance, but you can compute the free energies in other frames, in the same way that in cyber-witten theory, you can compute the free energies in the electric frame, or you can go to the monopole frame, or so-called magnetic frame, and so on. And in this talk, I would actually like to focus on a slightly different frame, which is called the magnetic frame. In this frame, which is the frame which is appropriate to analyzing the Calabria near the conifold point, the free energies have a slightly different structure. They look as follows. First of all, they are not fancying of the Keller parameter, but of another period, which is called a Hirlanda, and it's the vanishing period at the conifold point. And then they are given by a pole part, a pole for the 2G-2 with non-coefficient. Notice that this is the genus G-free energy of the SQL1 string. And then you have a power series in lambda, which can be computed by using different techniques. And I want to emphasize that this genus G-magnetic free energy has exactly the same information as the Gromov-witten free energies at large radius. It's just a change of simplistic frame. Now, out of these genus G-free energies in the magnetic frame, I can compute again a total string theory free energy, which is going to be a sum over all different genera of these FG-landers times the string coupling constant to the power 2G-minus. So this is the formal divergent genus expansion that I want to focus on. This is also to be divergent. These guys FG-landers diverge like 2G factorial, so again you have a formal power series which is in this one. Now if you want to start working with this power series, the first thing you have to do is to try to get as many terms as possible. For example, if you see historically how people started understanding the anharmonic oscillator, the quartic oscillator I told you, a very important thing was the work of Bender and Boo. So the best on the quartic oscillator. The first thing that Bender and Boo did in order to understand the quartic oscillator was to generate an old computer at the end of the 60s to generate the first 70 tens of the perturbation theory for this quartic oscillator. Now it looks like a joke. This takes three lines in mathematics and maybe a few seconds. But at that time it was hard and they did it. They really look at all these coefficients and they started studying how they diverge and so on. So this was a very important stepping and understanding, the non-pertile destroyer of the quartic oscillator. Now if you want to do something similar in topology history, the first thing you have to do is to try to generate as many FGs as you can. And actually one person has pushed this direction very much as I've reclaimed here and thanks to his techniques, which were by the way based on the Olamorphic and Mali questions of Bershasky, Cechoti, and Baffa, you can push this calculation for the local case to very high genus. So for example, in some of the calculations I'm going to show you today, we push this to genus 114. This, you know, to average this is not very impressive because he can get this with one minute of his laptop. But it's a lot of terms. It's a lot of terms. You cannot compute as many terms in many quantum series. Yes? Yes, no? Yeah, yeah, that's how you compute it. Yeah. It's very hard because they are functions. They are not numbers. You see, they were numbers. That would be fast. But these are functions. So you have to use recurrence for functions. And this is another world. For example, when you have functions, the recursion can involve derivatives. Right? And then, you know, this makes life slightly more complicated and the recursion can involve derivatives. Okay. Now, what we want to do now is to propose an unperturbed definition for this total free energy in this conical frame. I want to find a function such that it's well defined and that a synthetic expansion gives exactly all this genus information. Some time ago it was proposed that one way to get some unperturbed definition of these theories involves quantizing the mirror curve. So the mirror curve, as I was telling you before, is this function e to the x plus e to the y, blah blah blah, in many cases. You can write it very explicitly. And the idea was that you can get an unperturbed view of quantum topological strings by quantizing these mirror curves. So let me tell you exactly how you do this because this is not very simple. So the idea is that you take this curve that I wrote you before, this Newton polygon of the Tori-Kalabillau, and then the variables x and y, you are going to promote it to canonically commuting Heisenberg operators. So I declare that x and y are now operators from L to R which commute to i h bar. And h bar in this talk will be a real positive number as in the real world. Now for simplicity I will focus on mirrors of genus 1, but the theory can be extended to any genus curve. Now if you now take this function that I was writing before here, if you try to quantize it, you will find that there are ordering issues in this term here. But there is a very natural prescription invented by Byle already in the very beginning of quantum mechanics that tells you that you have a function like this and you have to proceed. Well, you have a minimal quantization prescription that makes this a self-adjoint operator, and this is the quantization procedure that we are going to use. We are going to use by quantization of these curves. And then when you do this for local p2, you get an operator in which you simply promote the terms that appear in the curve to operators. So you get e to the x plus e to the y plus e to the minus x minus y, and one point of the prescription is that in the Newton-Polygon you don't look at it. You just remove it. You just left it behind. And the reason is because this inner point is going to actually give you the eigenvalue for this operator. So technically just take the points which are in the boundary of the Newton-Polygon, but not the inner point. So this gives you an operator and this operator has been installed, for example, by using WKB, but one natural thing you can ask is the following. What kind of operator is it? We have all kinds of operators. Operators will give you boundary states, operators will give you scattering states. So what kind of operator is this one? Now, we actually conjectured with Alba and Yasuyuki that these operators are actually as good as they can be in a spectral theory. And this was actually proved by Renat Kassaf and myself, and also using more spectral theory techniques by Arila Pepe, Lukashimer and Leon Tachtaya. And the result we found is that the inverse of this type of operators are operators on L2R, which are positive definite, and they are of trace class. What does it mean that they are of trace class? The fact that they are of trace class means, first of all, that they are compact. So they have a discrete spectrum, and the spectrum of this inverse of this operator I'm going to write it as e to the minus em, because they are positive. So these are all positive numbers, so I can always write them as exponentials. And these numbers can be computed numerically. Remember what I was telling you at the very beginning of my talk. If you want to have an operator with definition, you have to have something computable. You cannot have something mystical. And then, you know, this eigenvalue cn of this operator inverse of o x can be computed numerically. So for example, this is a computation when h bar is equal to 2 pi, and here they are. So this is very similar, actually, this kind of problem is very similar to the Schrodinger theory. So in Schrodinger theory, if your potential goes to infinity at infinity, you have only a discrete number of eigenvalues, and this is the simple situation you can have in operator theory, because then we know we have more complicated spectral problems, we have scattering states, or we have periodic potentials, but this is the simple situation which you can have, where you have just a discrete set of eigenvalues like in a confining potency inverse theory. The property of this of trace class means that the zeta functions, you want the spectral traces of this operator are all well defined. I'm going to explain this in the next transparency. So how do you encode the spectral information of a trace class operator? Well, the simplest thing you can do is to compute the spectral traces, which are if you want discrete analogues of the zeta function. So, just take the trace of row x to a non-neg... sorry, these zeroes wouldn't be here. You have to start with one, of course. So you take a non-negative integer, and you form all these sums, which are sums over the spectrum of this operator row x. And the property that this operator is trading class means that all these traces are well defined, are finite numbers. Now, this is one way in which you can encode the spectral properties of this operator through the spectral traces, but a much more elegant way involves what is called the Freddholm determinant. When the operator is trace class, an important result of the spectral theory says that you can compute this, you can define the spectral determinant as the infinite dimensional determinant of one plus kappa times row x. So kappa here is a parameter that I'm going to introduce. We'll see later on that it's actually related to the kappa parameter that appears in the spectral curve. It's actually the modulus of the spectral curve, and this determinant can be written in many ways. There are various theorems. One theorem expresses some sort of exponential of combination of these spectral traces, but one very important theorem in spectral theory tells me that the Freddholm determinant of a trace class operator is an entire function of kappa. So this guy turns out to be an entire function of kappa, and in particular, it can expand it around the origin. When I expand this function around the origin, I get coefficients with depends on h-bar and on n, and this object is going to be very important in our talk. I'm going to call it fermionic spectral trace. It's a natural logic that appears in spectral theory, and notice that through this identity you can compute all these fermionic spectral trace from the standard spectral traces that I defined here. So these are kind of basic ingredients of spectral theory. Now for those of you who are not familiar with spectral theory or who do not feel very comfortable with this, there is a much more physical way of thinking about this, which is the following. Think about rho as a density matrix, okay? A density matrix of a quantum system, and let's imagine that this is the spectrum of this density matrix. Now, in this picture, this Freddholm determinant is what you call the grand canonical partition function of an ensemble of fermions with fugacity, sorry, with fugacity kappa. So kappa is the fugacity of a Fermi gas, and this is interpreted as the grand canonical partition function. And then these fermionic spectral traces are the canonical partition functions for a gas of n fermions with that density matrix rho. So that's why I call it fermionic spectral traces because they are the canonical partition functions of an ensemble of n fermions. So this is the canonical ensemble and this is the grand canonical ensemble, okay? So you can reinterpret all these notions of spectral theory in terms of gases of free Fermi gases, okay? So here there is only this is the density matrix, the one particle density matrix. So these are like fermions in a single potential with these energy levels. And then you fill the energy levels using the exclusion principle and then you get these thermodynamic quantities. Okay, so these fermionic spectral traces are well defined. And they are well defined because I was very careful in checking that these operators are of trace class. And they can compute it from first principles in the quantum mechanical model. For example, if n is equal to 1, I can compute this number of the kernel of this operator row so the diagonal kernel, okay? This is an elementary formula that falls from the formula that I was showing you before. And this integral converges precisely because row is of trace class. Now it turns out that for these operators that come from mirror curves, all these operators, the integral kernels, the kernels of these operators can be written explicitly for many geometries. And then they involve what is called the Fadeyev quantum dialogarism. And this was shown in a paper by Casayar, and actually this means that you can very often compute these numbers analytically. And this is a rare luxury in quantum mechanics. If I give you a generic confining point of view in quantum mechanics and I ask you to compute the spectral trace you will have a hard time because it's not so easy to find these integral kernels explicitly. For example, André Bogos completed some time ago the first spectral trace for the quartic oscillator. So he actually managed to compute these but already the second spectral trace when n is equal to 2, it's very very hard to compute. While in these cases sometimes we can compute these to very high order and actually there was a paper by Masajito, my former student Pavel Putro, where they did a similar computation in a similar contest and they were able to get many of these fermionic spectral traces explicitly. So this is a rare luxury in quantum mechanics. And what is remarkable in a sense one remarkable spin-off of this theory that I'm presenting you here is that these operators are much much simpler than any operator that you find in Schrodinger's theory. These operators are such that you can very often write down explicit quantization conditions, explicit formula for the Freckland determinants and things that you cannot even dream of in Schrodinger's theory. So in a sense exponentiating the Heisenberg barrier the Heisenberg barrier was linked to a spectral theory which is much nicer than the usual spectral theory which use x and p. So you have to take e to the x and e to the p and then suddenly these operators become much more manageable than the standard operators that you find in quantum mechanics. And one of the nice things that you can do with all this theory is actually write down explicit expressions for example for the influences of these operators and for the spectrum which are really very very hard to to find in any non-trivial confining quantum potential in quantum mechanics. So of course the harmonic oscillator. Okay, as I said this this can be computed very explicitly in some cases and the quantum theory is particularly simple when h bar is equal to 2 pi. More generally when h bar is a rational multiple of pi this theory can be computed in a very nice way and the reason is technical and is that the object which is behind these guys is the quantum dialogarism. And it's known that the Fadellius quantum dialogarism simplifies enormously when the parameter is rational. This was found very recently by Garouf-Falidis and Kashaev that essentially when the parameter of the quantum dialogarism becomes rational the quantum dialogarism which is a complicated function collapses to something simpler and this is actually the reason why for these values of h bar these become much simpler. Now you can actually just to show you that this can be computed very explicitly I showed you here some values for local p2 so for example when h bar is equal to 1 and h bar is equal to 2 pi which have 1 ninth when h bar is equal to 4 pi and equal to you get a more complicated number but still something that you can write down and it's not a transcendental it's not really very transcendental you have a bit of irrationality here but it's essentially up to this square root of 3 these are rational numbers and pi. Well now my main claim is the following let's think about this as a partition function depending on n and h bar and let's take its log to have the free energy I claim that this guy gives an unperturbed definition of the topological stream free energy in the conical frame that's my main claim so if I give you the definition of an unperturbed definition I have to make sure that this satisfies the criteria that I told you well first of all I need a dictionary relating the parameters here to the parameters in string theory what are the parameters in string theory in constant and the conical modulus which corresponds if you want to the to the radius in an ADS compactification now the dictionary is the following the string coupling constant is the inverse of the plan constant so this is already very interesting because it's telling you something about this definition it's telling you that this is a strong weak coupling definition the sense that when the quantum mechanical problem is weakly coupled the string theory will be strongly coupled and vice versa the second definition I need is the modulus of the string theory the conical modulus as a function of n and h bar and this is the dictionary so lambda is actually the top parameter for a theory with rank n and coupling constant 1 over h bar ok now does this nonperturbed definition agree with the requirements? well the first requirement is that this guy is well defined but this is exactly what we have shown we have proved we have proved that the operators are trace class so all these spectral matrices are well defined for any positive integer n and any real and positive integer so this is a well defined object now the second thing we have to check is the asymptotic property that this object has an asymptotic expansion which reproduces the topological string inverse expansion and nothing else the asymptotic I have to consider is the standard top limit so this is very much in the speed of ADS-CFT dualities so you take n to infinity you will take h bar to infinity because you want gs going to 0 and such that the top coupling is fixed and then we conjecture that the asymptotic expansion of this guy is exactly giving you the genus expansion of the topological string at the conical frame now this is the part of my proposal which is conjectural so of course you know this is a conjecture but conjectures if they are well post can be checked and actually this asymptotic expansion has been checked in enormous detail there are all kinds of checks for example there is a very nice check which is really a direct check which is just take you know set up and compute as Feynman used to say just take this guy and you span it and you find the expansion and the reason you can do this is that as I said this kernel is very explicitly so you can sometimes write formulae for these z's as matrix integrals and then what you find is that this expansion is just the top the expansion of the matrix integral it's a matrix integral whose integrand involves the quantum dialog of in a very crucial way now when you actually expand this matrix integral you can check this really analytically coefficient by coefficient and this is something that we did and then we really check it you know of course it's a check, we cannot prove this for all genera and so on but we can check that this is exactly as predicted I will give you some additional indirect evidence for this in the talk that actually this guy has a asymptotic expansion this topological string genus free energies in the conifers ok now we can address the issue of the discreteness of n ok this is always like a problem for all these large n dualities with case series because n is discrete or it is continuous what is n and so on now as I was mentioning before here when I do in this dictionary you see n is an integer and so you see that I cannot cover arbitrary real gs and arbitrary real lambda by using just an integer n so you can ask what happens with asymptotics with n is not an integer and so on well in this case we are very lucky because we have even a stronger conjecture that gives an exact formula for this for this fermionic spectral trace in terms of a function that can be computed from the vps invariance of the calabia of x I don't have time unfortunately to explain you this formula in detail I think Alba in the here talking the afternoon will explain you a little bit more about this but I only want to say that this is an exact formula it's not an asymptotic formula and the asymptotics that I was presenting before follows from this stronger conjecture now the nice thing about this conjecture is that it presents fermionic spectral trace and some sort of Laplace transform of this function here and you can actually see that if this formula is true then you can use it to extend analytically this fermionic spectral trace to an entire function of the complex n plane so our conjecture not only reproduces the values of this fermionic spectral trace but actually interpolates on the full complex plane so in a sense there is no problem with n being an integer because we can extend it to a function in the complex plane this is something that you can do if you assume this conjecture but if this conjecture is true this means something very interesting this means that these fermionic spectral traces of the variable n and this is not so crazy because this is exactly what you do with the spectral zeta function the spectral zeta function you define it for some values of the you know you take the sum of the spectrum like this maybe let me write this so you have something like this right assuming some simple and then you know when s is equal to an integer l this is precisely the trace of rho to the l that I was defining before but you know that this function can be sometimes extended to the full complex plane with some singularities and so on what I'm saying is something very similar I'm saying that this fermionic spectral trace which is a spectral trace after all can be extended to be a function defined on the full complex plane there are functional equations that you can write actually for this guy it's not so well but for the generating function which is the freckling determinant the freckling determinant satisfies some different equations that you can use sometimes but you know this is not so important for us because we can actually write exactly with this conjecture so in a paper we wrote with Alvin Jasnicki we explored some of these functional equations for the freckling determinant but we didn't really push you very much this direction okay so this is this is my this is my number to the definition it satisfies the criteria I proposed myself at the very beginning on my top there is still a conjectural part which is that the asymptotics is actually the topical stream but we have checked it and we have no doubt that this works now what I want to do is to make contact with something much more traditional which is the theory of Borel Resumation which has been now re-bumped as the theory of Resurgence okay so how would you pose this connection well let's go back to my problem of the quartic oscillator I gave you this Pertorative series at the very beginning but then you know you don't like to work very hard on spectral theory I mean this series I can just do some sort of resumation on the exact energy levels and you know people in physics have developed resumation techniques that given a Pertorative series you can resum it in a clever way because this series is divergent so you cannot resum it in a trivial way but there are clever ways maybe to resum it so I get a number at the end of the day and this is going to be my non-pertorative number okay so these techniques go by the name of Borel Resumation or the theory of Resurgence and so on and now since I have proposed to you a non-pertorative definition of topological stream you know you can ask okay how this is connected to the theory of Borel Resumation because after all in many cases you can do these resumations what's the relationship between these two things so as I said there is a traditional way to produce well defined quantities from factorial divergence series and this is called the theory of Borel Resumation so how does this work in a cartoonish way in my that I give you a formal power series in some variable G such that the coefficients are very diverse factorial this means that the leading order of A n is n factorial and then you have a sub leading exponential growth characterizing these coefficients so of course this series does not converge because A n grows like n factorial so a very easy way to kind of cut these diverges is to divide by n factorial and this is what the Borel transform does the Borel transform takes this formal power series and gives me another formal power series which is obtained by dividing A n by n factorial now typically the series that you are thinking this way has a final radius of converges around the origin and now what you can try to do is this function which is well defined in a neighborhood of the origin try to analytically extend it to the positive real axis and do a Laplace transform in this way so you take this function you multiply z by z you do a Laplace transform and then you get a function of z now this function if it exists is very easy to see that this asymptotic expansion is the original series that you started with so this is a fantastic machine which turns asymptotic expansions into well-defined functions provided some conditions are fulfilled now I gave you a very sketchy story of how this works you want to know more details you can learn them in my book I will continue to make some publicity of my recent book where all this is explained with examples and so on so this procedure does not always make sense because typically when you do this resumation when you do this model transform the analytic structure of this guy involves some singularities now the singularities are not in the positive real axis which is where you do the Laplace transform everything is more or less okay but if you have a singularity then the whole procedure is abstracted and you say that the series is not Borel's sample now in practice if I give you a crazy series how do you do this? well you have to rely on numerics very often so you have to check the absence of singularities and perform the resumation but usually if you are lucky you can see for example that no singularities will accumulate in the positive real axis or you will see them and so on so many of these things can be checked numerically now once you know that there is this procedure to make sense of of perturative series as the series the question now is the following imagine that I have a quantum theory and I have one of these divergent series, okay now imagine that I have given you a non-perturative definition of what this perturbed series stands for, for example in the case of the quartic oscillator I gave you a formal power series and I also gave you the non-particle definition based on spectral theory of this potential how are these two things related? how is the Borel resumation of this formal perturative series related to the non-perturbed definition that I gave you are these equal? are they different? this is a classical question in the theory of Borel resumation because Borel resumation if the series is Borel summed always gives you some number the question is this number is the one that you want maybe it's not the one that you want and the most of the arguments of the theory of Borel resumation are actually based on giving you conditions for these things to work okay now if the series is Borel summed then if all this procedure of Borel resumation can be actually be done without abstractions then you know then you get a number and the question is how is this number then related to original quantity now in the case of the quartic oscillator it was an important result in mathematical thesis in the 70s that these two numbers agree the Borel resumation first of all the series that you obtain for the Granthus Detener is Borel summed and the resumation the Borel resumation of this function gives you a well defined function of the coupling and it turns out to be exactly equal to the Granthus state energy that you find using operator theory so this was very good this is a classical result which tells you in the quartic oscillator you don't have to worry having the perturative series and having the exact answer the number definition is more or less the same because there is no extra stuff you do Borel resumation and you recover your original quantity but you know this is not the rule it may happen that the series is Borel summed and still Borel resumation does not give you the quantity you want and this sounds like a surprise for many people because usually you think that for this series not to work you have to have a series which is not Borel summed but it was already realized by Valiant, Parisi and Bogos some time ago and then rediscovered in a sense by Alba, Myself and Sapols in some more modern examples that isn't always the case you can have a series which is Borel summed and still this Borel resumation does not recover the original quantity and we gave some criteria for when this happens and so on now in some cases it's even worse the series is not even Borel summed so what do you do you have for example singularities on the posterior axis you cannot really define the Borel the Laplace transform and then things do not work and this is actually in a sense the default situation in physics for example you take the double well potential in quantum mechanics and you compute the Borel series well it's not Borel summed you cannot go through all this procedure in a naive way, you have to do something else so what you have to do when this is the case well you need additional information it's clear that the exact answer cannot be given just in terms of the you need something else and this is what in physics we call a non-pertorative effect yes this is the quartic oscillator in quantum mechanics the quartic oscillator but the pure quartic oscillator in the pure quartic oscillator it happens that no no no, it's not true it's not true, the Borel resumation this is why this was a surprise usually people say the series is Borel resumable and done, no, it's not true it's not true and these guys show it in a quantum mechanical example and we actually this is going to be the situation in the political history and it was the situation in the one over an expansion so and this is because this is very subtle phenomenon because it's associated to the appearance of complex instantons instantons with complex action when the instantons have complex action they're you know you have a single light in the Borel plane not in the positive real axis but they still can contribute to the as a non-pertorative effect so it's a very subtle thing but it happens so what is a non-pertorative effect? well this is again, it's also one of these words that is used all the time but fortunately we have a kind of more formal framework to understand this which is the theory of transience now very roughly I cannot go into all these details because this involves all these theory of resonance and so on but very roughly what happens is the following in a quantum theory if you perturb around the trivial vacuum around the trivial saddle point of the path integral you find the standard perturbative series if you do the perturbative expansion around a non-trivial saddle or an instanton you find a transidious so a transidious is just a formal sum of formal series so you start with the perturbative series which is the expansion on the trivial saddle and now you do a perturbation around non-trivial saddle a non-trivial saddle will have a non-trivial action which appears typically as e to the minus a over g where g is the coupling constant and then when you do loops around the instanton you will get another formal series again in the power in the coupling constant g and then what you do to form a transidious you add this with the series that you find after spanning around a one instanton but you can have multi-instantons in the theory so you have to add them as well and so on and the instanton contribution is multi-private by a parameter which actually is ambiguous and so we leave it as a parameter and the reason why it is ambiguous is because as I said all these series are formal so you don't really can make sense of this series unless you do something very concrete and this means that there will be always a parameter reflecting this ambiguity from the fact that these series are purely formal now it turns out that you can actually again and skipping here many things but it turns out that even when the theory is not Borel summable if you include the transidious there is a formal procedure to do Borel resumation even taking into account the singularities which is called sometimes Borel ecal resumation and essentially what happens is that instead of doing the integral around the positive real axis you go around the singularities through a appropriate path in the complex plane and then you get something which is ambiguous and complex but this is going to be cured by the transidious effect ok it's a complicated theory I'm not going to get into the details but at the end of the day the theory of Borel ecal tells me that if I give you a perturative series plus all these instanton corrections I can compute numbers by doing Borel ecal resumation but these numbers will depend on parameters and these parameters are associated to the instantons that appear in the theory ok so if you are if you want to pay the price of having an ambiguity in your theory you can do this instanton stuff so sometimes people say that resurgence techniques give you a number this is false never buy this because resurgence always comes with parameters so it's not true that you get an ambiguous answer you get ambiguous answers and actually this was and here in Italy I want to say that many of these things were discovered by Parisi while studying you know renormal effects in QCD and so on he realized very quickly that the coefficient appearing in front of a number 35 in a quantum theory is intrinsically ambiguous so you cannot fix it unless you give me some extra information so these techniques of Borel resumation give me multi-parameter families of non-perturative objects fully defined yes exactly so what I'm saying is for example here in the WL potential if you do Borel resumation there will be an ambiguity how do you fix this ambiguity by going back to your non-perturative definition of what is a ground state energy using operator theory I mean Schrodinger theory tells you that in this system there is a single ground state energy because this is a confining potential so it's going to have a ground state energy now if you want to calculate this ground state energy using this technique oops you will have an ambiguity here and this you will have to fix at the end of the day okay you will have to fix by comparing to non-perturative definition so that's why Resorges doesn't give you non-ambiguous non-perturative definitions it encodes the ambiguity of non-perturative definition in a finite set of numbers but it's still ambiguous okay so now we can ask again now that we have this enlarged framework can we reconstruct a non-perturative function from a trans-series not just the perturative series but the series plus all these terms of corrections and this is what I'm going to call the semi-classical decoding conjecture it's not always the case that you can do it so the semi-classical decoding conjecture says that non-perturative functions in quantum theory can be written as the Borelical resumation of a trans-series and this is true in quantum mechanics in many examples in quantum mechanics it's true in some simple systems in some simple quantum field theories in long dimensions or topological or for example in supersymmetric systems but it's probably not true in Jamil's theory in fluid volume so you cannot decode the ground state properties of Jamil's theory using semi-classical objects which is the intuition which is behind this semi-classical decoding conjecture ok, so this was the tour through through Resolvency's theory a little bit quick so let me go back to my my original problem with non-perturative topological strings so I have given you this non-perturative definition of topological strings now the question is is the semi-classical decoding conjecture true for this? can I decode this non-perturative definition in terms of some perturative series plus some instanton corrections now the first thing to do if you want to do this first of all you have to see if your perturative series is borel-sammable or not that's the first thing, because if it is borel-sammable then maybe life is simple and just by doing borel-resumption you are done so this is the first thing you have to do and this is something that we could have done long ago we could have long ago look at all these genus G-free energies of topological theory and ask if they are if the theory is borel-sammable or not and in the following just for concreteness I will focus on local p2 now it turns out that the perturative genus expansion is borel-sammable for almost all real and positive values of lambda the way you do it is to actually compute this borel transform numerically and see there are poles on the positive real axis stable poles so you see that here after 114 genus general analysis we don't have any abstraction on the positive real axis we see that we have poles in the complex plane and the binary line so these are sources for non-perturative effects but they are not on the positive real axis so you can't do a standard borel-resumption of the topological stream you can just take your genus expansion in the conifold frame as for example Abreff computed like few years ago and then you push the computer and then try to do borel-resumption so you can get numerical answers for this so how do they compare the definition I gave you and this is really, here we are doing precision numerics of non-perturbe topological stream we are really able to get to numbers and that's why I think this proposal is very concrete yes, actually for example these poles that I put here with lines with these circles these are the the value of these poles is a period of the Calabria and this is a nice interesting property that the singularities in the borel planes seem to be always combinations of the periods of the Calabria so for example this is one of the periods, this is another one and then you have shifts in in 4 pi square, you see so there is a very nice strata which we still don't fully understand but in a sense if you want periods measure the mass of d-brains, so in a sense these actions are masses of d-brains topological d-brains which are non-perturbe structures, I will come back to this at the end of this sorry, this is the complex lambda plane yes, yes so there is, and this is for example, you see that you have lambda and then you have the complex conjugate right? you have lambda minus lambda and then you have the complex conjugate and this is because when lambda is real, the perturbe series is real, and then if you have a complex object you have to have this conjugate and also because the power series is in even powers of g-strange you will have to have plus lambda and minus plus a singularity and minus a singularity so all these symmetries are at the consequence of the structure, the reality and parity structure of the perturbe series okay, oops okay, so let's do the spinning let me take the Borel resumption of the perturbe series when n is equal to 2 and h is equal to 4 pi so I can compute with my non-perturbed definition so what I get is this number here and all these digits here are stable digits okay, so this I could have done like many years ago now our non-perturbed definition gives remember I was quoting this number for you phi over 324 minus 1 over square root of 3 times p times 12 and this number you write it in digits is this one so the first thing you notice are incredibly close and this is again another way of seeing that our non-perturbed definition is actually on the right track in the sense that it's almost the Borel resumption of the original series so few years ago we could have done this calculation we would have obtained this number here and what is amazing is that now we have an approximation to this number which is this spectral trace so you know this spectral theory knows something about the topological stream and this is actually reflecting the fact reflecting that our conjecture is true reflecting that the fact that our conjecture predicts that the asymptotic expansion of these numbers has to be the perturbed genus expansion of the topological stream and this is one way of seeing it so these numbers are incredibly close but they are not exactly the same so we conclude that we are in the second case I was presenting you before the theory is Borel summable it does not reproduce the Borel resumption that does not reproduce our non-perturbed definition and this we have seen before in this example of Vali and Parisi and Boros but in a closer example it was shown by Alba Szabolzak and myself and this is exactly what happens you look at the one over an expansion of the ABJM matrix model in ABJM matrix model you can compute the matrix model for finite values of n you can compute the one over an expansion and it's Borel summable both of them and this is a mismatch due to the fact that the perturbed one over an expansion is not including all non-perturbed and this is exactly the same our non-perturbed definition is not exactly agreeing with this Borel resumption because we are missing non-perturbed effects here now you can do this experiment for many values of lambda the agreement is actually quite amazing so here I take h-value equal to 4 pi and vary n so you see I can do integer values of n and I can do also non-integer values of n assuming using this formula that I was presenting you before so you see the the black line is the Borel resumption and the dots are the values that we obtain from the spectral theory from quantum mechanical operator coming from the mirror course so you see the difference is not visible to the naked eye so all this Borel resumption is always very close to the non-perturbed definition we get and this is exactly what you should expect from our conjecture because if our conjecture is true the Borel resumption of the perturbed series and our non-perturbed definition should only differ in exponentially small effects in effects which are not visible in an asymptotic response now the question is can we compute this effect explicitly can we actually find a transidious that this Borel decal resumption gives back the exact answer and of course this is a very complicated problem in stream theory because in stream theory already computing the perturbed effect the perturbed series is very hard computing non-perturbed effect in stream theory is even not a well defined problem because you don't have a stream field theory from which you can define this so how can you do this in topological streams how can you compute non-perturbed effects the inspiration here comes from the theory of matrix models now in the theory of matrix models in the theory of matrix models you have the matrix integral and the matrix integral in principle you can compute instanton effects from the matrix integral but in the theory of matrix models you also have recursion relations and then I come back to your observation there are recursion relations that determine the perturative series and very typically you can promote these recursion relations that also compute exponentially small quantities and we have a recursion relations for the genus G free energies and these are the holomorphic and amalic questions of Bersasque Checotiburia math so this was an idea proposed by Ricardo Cosso Santamaria, Ricardo Esquiapa and collaborators which is ok and take VCOV which is a recursion relation for the perturbed free energies and now I assume that instead of plugging in this recursion of the perturative series I'm going to plug also a non-perturative effect like this and then I will try to solve for this of course there will be holomorphic ambiguities and so on you have to fix them and so on but now I'm going to be a little bit fast because the technical details are actually complicated but at the end of the day you can solve for these non-perturative corrections using VCOV in terms of perturative topological data and assuming that the instanton actions this is an assumption and assuming that the instanton actions are given by periods of the calabria this is a checkable assumption because everything that you actually write down here can be tested through the well known mechanism of large order of perturbation theory Bender and Boo found that whenever you have non-perturved correction a non-perturved trans-serious effect in your theory every information here instead of this coefficient which is ambiguous can be read from the large order behavior of perturbation theory and this is something that is very important so the fact that these instanton actions are periods of the calabria can be checked against the large order behavior of these series so all these things are testable and then I'm skipping for you the details so let me just show you one number to show you that this more or less works so what we did at the end of the day was to include the one instanton correction given by this procedure we put a fixed parameter c which we fixed actually a quite simple value of c so it turns out to depend on n but it's just this thing here so this is the value of c that we this is a value that has to be fixed experimentally we don't have any clue of why this works but this is the universal value we use and then you see that now remember this was the Borel resumption of the perturbed series, we matched some digits this is the exact answer and now if we add the one instanton correction we improve this precision to all the precision that is given by a one instanton correction of course if you want to improve even more this and so on so of course we did this for many many values of lambda and in all cases it works so this is evidence that our non-perturbed completion which is a well defined definition of topography-gastry theory not only is a well defined definition but it can also decode semi-classically in terms of a trans series so at the end of the day we have provided essentially everything we needed we have provided a non-perturbed definition this non-perturbed definition contains non-perturbed sectors which can be formally computed through a modification of the holomorphic anomaly equations and of course the question now that is mainly open is how would you interpret these non-perturbed effects, I mean maybe they are due to topological de-brains they look like they are due to topological de-brains but of course it's very hard to define a first principle calculus of topological de-brains so that's why we use the BCOV holomorphic anomaly equations as a way to get this indirectly ok, let me conclude here so more or less in time I hope you are not, 5 minutes later so we have given a rigorous and concrete non-perturbed definition of topological de-brains that's for me the most important result and this definition is interesting because the question here is not who gives the non-perturbed definition it's very easy to give non-perturbed definitions just by doable resumption for example the interesting thing of our definition is it's non-trivial it relates the topological string theory to a spectral theory and we also have shown that also this definition is highly non-trivial it can't be decoded through the standard theory of resorience in terms of perturbative series plus trans series effects and so on now there are many questions that you can ask for example as I was mentioning this quantity can be written as a matrix model so in principle you could see that the instanton effects can be completed directly from this matrix model but this is a very tricky calculation because this matrix model involves this fadejf quantum dialog and so it's a complicated thing now, as I said there might be other non-perturbed definitions we are not claiming that this is the best one it's definitely one non-perturbed definition it's interesting because it makes this connection to spectral theory and it has given many many interesting results already on Cala Villaus even sometimes this non-perturbed definition gives perturbed information that was not known before for example it gives you for free the values of the periods at the conical point which is a well known problem in number theoretical properties of Cala Villaus that can be solved here with operator theory so we can use operator theory to solve the classical problem of a special geometry of Cala Villaus of course, you know, the main thing is how would you prove this correspondence how would you prove this connection between spectral theory and topological strings because my talk was based on one conjecture which is that the asymptotic expansion of these objects is the standard topological string this is part of the conjectures we have and there is a very interesting particular case of this conjecture which was proved recently here at Trieste by Julio, Alessandro and Alva and Alva is going to talk about this in the afternoon session and of course we would like to have a general proof and this is really a well defined mathematical problem we are making a statement of the spectral theory of song operators and we are relating this to a well defined and numerative theory of Cala Villaus so these two things are deeply related and I think this is a very interesting correspondence and I hope mathematicians will also do some work on this and maybe they will be giving us a proof in the near future thank you