 OK, zato sem zelo podeljno površen, da je to še začala. V tem vseh, da sem podeljno podeljno površen, da se možeš odvršen? Ako se možeš odvršen? OK, zato sem podeljno podeljno površen, da se možeš odvršen, da se možeš odvršen? in v tem, da je to in ideja zahovoril. Zelo začetno je, da bomo začetno besetite povedanje, a to je vse tega, kako je načinilo v koraboraciju s Marcos Marino, Jace Yukyatsuda. I pri druhej seče, da bomo predstaviti načinit, ta formulacija delala za sabervitanje teori. Tama je dobro v teori, ko je v Rešim Jovgeni Monelli in Sandro Tanze. OK, z nekaj nekaj generičnih počkev, v težnih pravamo v duvalitih, in v nekaj duvalitih, kaj je zelo v nekaj način, nekaj geometry, strejk teori in graveti. In v nekaj način je operator teori, matrič model, in kvantomekaniki. we have two quite different theories which are related by means of these dualities. In this talk I'm not going to discuss in detail general property of dualities but I will more focus on some concrete examples. An example that maybe you are familiar with is width and contricature. Zelo jazne možno je vzgledaj po moru zelo naprejveno gravidje in je odličena na vsevsevstvene teori v moduče vzgleda. Bidem je dole možnosti minimali, ki je odličeno na matrički model. In je to vzgleda, da je vzgleda vsevstvene teori, in tudi teori, da ima bojne zrednje izgleda vse površčenje, zato je zelo površčenje 1. Zelo to je nekaj izgled, nije boš se na to, da je izgleda in vzelo, zelo je, da boš se vzelo, da je bilo začala površčenje koča, zato je bilo začala, da je bilo začala, zelo je bilo začala, da je bilo začala, Ali zelo, da jih so prišličeni, da je to hradnath trenikov, da ja razumelam, ki so v potenetih umedi, in z druha rezova, da je to najbolj. Tudi početni je, da imesem vseh nekij udelov in medželje in matematiki, ki svečem v porej odaj, in z njega se z tudi opravil Vojstina, do boh, je zelo več svobodno, da se vse načo načo je zelo. Načo je zelo, da je zelo vse vse vzelo in vzelo in vzelo in tudi vzelo in vzelo. Vse je zelo, da se vse vzelo in vzelo in vzelo in vzelo. Vzelo in vzelo in načo je zelo, in tudi vzelo in vzelo. Zato se boh je bolj mladosti, da se vzelo in theo zelo je srečen, ker jih mene začeli, da bi se neko prepenetilni pred včetni, neko neko relativ galaxiesi teori, kaj stejne teori, v tijelijkej prjisti teori, ki je sreko mekanikulne teori, z vsim Tafela. In v neko v sejvali k Resilcij s drzvami, je vse tudi adiaziprodjosti, ješko se neko prepenetilni pred včetni, z soustejne teori, kot vzeljno iz glasbo, v tem, da je vsežitak in si inje glasbo, in zelo je tudi dobrozvodno, da je vsežitak in začalj na konording, ker je tegne, da nekaj je dobrozvodno, da nekaj je nekaj vsežitak, zelo, na različne, vade matriče. Zelo več, da nekaj smel. Vzelo je tudi to, da je to, bilo tudi to, tudi to, da je to, da je to, da je to, je to, da Markos je vsega izgleda. V stvari, kaj je to teori potopoločnico, v sej manjifoldu, ko je tori, in je matematik, ko je kateropart, je to zelo inumerativ. V drugi stari, je konto mechanik, in in idej, fermigaz. And these correspond to operator or spectre theory. Okay, so roughly speaking, this is the plan of the talk. I will start by reviewing some aspect of this duality, but I will need later. But then I will focus on a particular limit of this duality, where things simplify in a certain sense. And then you obtain a new duality, which rate cyberwitten theory in four dimension. And on the other side you have, again, an another ideal Fermigas in an external potential. And what is nice about this level here, is that actually you can prove this duality, because the main object that characterizes this theory and this theory, they both satisfy the same differential equation, which is a Pan-Levé equation. And to show that in this side, we have to use some recently recent work by Lissavi and collaborator, also Misha, which Kela talked just before. And on the other side, instead, we have to use some a little bit more older results by Zamalotikov and other people. Okay, so I can start by reviewing a little bit some aspect of this duality. So as I was saying, on one side, we have topological string theory. So this is a theory, which studies some maps that goes from a Riemann surface of genus G into a certain target space. Now, in this talk, I will always focus on the case in which the target space is a Kalabija manifold, which is called local p1 times c1. So this is constructed by taking two copies of p1, and then you put a bundle. And physically, these maps, they describe the trajectory of your string in the background geometry. And at the perturbative level, as Marcus was mentioning in the morning, this theory is characterized by what are called the genus G-free energy, which are this quantity here. So t is like a geometrical parameter which is related to the size of this background geometry. And this genus G-free energy basically counts in some sense the number of this type of maps. And then what you do usually is that you organize this genus G-free energy into a common generating function which is called the Gopakumavafa free energy, and is this object here. So Gs is what is called the string coupling. It gives the strength of the interaction of the string in spacetime. Many of the sum, these here are numbers, they are called the Gopakumavafa invariant, and they are related to the counting of these maps. And then you have this piece here, and here you have e to the keller parameter. Now as Clem was also mentioning yesterday, so if you take for instance G equal to zero, here you have one over sine square. So every time this is like a multiple of pi, you get a pole. And here you have a summation, so you have a dense set of poles along the real axis. So this function is somehow problematic when you restrict yourself to the real axis. So what you can actually do is that you combine this quantity with something else in such a way that at the end all this pole cancels, and you end up with quantity, that is what we call the grand potential, which is defining this way, so you take this and then you add some extra piece. And this summation is done in such a way that at the end this grand potential here is going to be well defined for every value of the coupling. And this extra piece also is also a known quantity. It's called the Necrozov-Shatashvili free energy. It's something that people know how to compute if you work in topological string. And it also has an explicit expression in term of some invariant that characterizes the underline geometry. Now I'm not going to give you all the details because we don't need them. Just remember that we have a new quantity, which is the grand potential, and this is defined for every value of the coupling, and we can really write down an explicit expression for this quantity in term of an invariant. And then what we do is that we take this grand potential, so this depends on two parameters. One is the string coupling, and the other one is this parameter mu, which is related to kappa in this way here. And this is like a geometrical parameter, which is basically the complex model associated with the underline geometry. And then what you do is, you take this grand potential, you shift by 2 pi i, and then you sum over all the shift. So in this way you obtain an object, which we call the grand canonical partition function of topological string, and this object is manifestally invariant under a shift of mu by 2 pi i. And, okay, so this is an important quantity, and so remember that because they will come back later in the talk. So now if we go to the other side of the duality, the underline physical theory here is an ideal fermi gas in an external potential. Now, when you have this kind of gas, everything is determined when you have the density matrix. So according to this duality, the density matrix that characterizes the gas, which is dual to topological string on local p1 times c1, is the one whose density matrix is constructed by quantizing the mirror curve as Marcus was mentioning this morning. In the case of this geometry, the local p1 times c1 that I was mentioning before, the operator that you get is this one here. So m is a parameter. We are going to take this positive. And, yeah, okay. So basically this is the density, which describes the fermi gas, which is dual to topological string on local p1 times c1 according to our duality. So this is a discrete spectrum, the rho is a trace class operator and it's precisely the kind of operator that Marcus was mentioning this morning. And then what you do is that basically you define the grand canonical partition function of this operator in this way. So this is what you do when you study fermi gas in statistical mechanics. So it's the determinant of 1 plus kappa rho. Kappa plays the role of the fugacity of your gas. And then this you can write it as a product over all the energy levels. So e to the en is the eigenvalue of this operator. And then if you do expansion around the small value of the fugacity, the coefficient correspond to what is called the canonical partition function of your gas. And one more comment that I will need later is that if you have an explicit expression for this spectral determinant, then you can compute the spectrum of this operator simply by looking at the vanishing locus. Because this vanishes precisely when kappa is given by the energy level. So now that we have seen the two sides of the duality, basically this is like the global picture. We have topological string theory on this geometry. And the topological string partition function compute the numerative invariant for this geometry here. And then there is mirror symmetry which basically relate this geometry to this geometry here. And this is described by this curve here. And then you can quantize this curve by promoting x and p to some operator that fulfill the commutation relation. And then you get this operator and you think of this operator as the density matrix of vanity alpha omega. And then what this duality is telling you, basically is that this theory and this theory are really closely related. So this is just like at the level of a picture. But actually you can make all this very precise and you can make very precise statement for this identification. So this duality can be summarized into precise statement. One is the statement about the grand canonical partition function and the other one is a statement about the canonical partition function. Just to summarize a bit. So at the level of the grand canonical partition function the conjecture, the duality can be encoded in this conjecture here. So on this side you have the grand canonical partition function of topological string and in this side here is the spectral determinant of your gas. So here mu is like a geometrical parameter that is related to the complex model, as I was saying. And in this side instead it plays the role of the chemical potential. And then the string coupling constant is related to the inverse of the plant constant in the gas. So in that sense this give you an exact relation between enumerative geometry on this side, because this compute the enumerative invariant of the background geometry and spectral theory. Because basically this is what contains the spectral information on this operator here. So in that sense this duality give you a new family of exactly solvable problem in spectral theory. So a new family of operator for which you can write down an exact closest version of the quantization condition and the spectral determinant also that determines the spectrum of this operator. And here we have a concrete example. So this is the operator that you get when you quantize the mirror curve to local p1 times p1. If you choose the commutation relation in this way so h bar is equal to 2 pi then you know that the spectrum of this operator is given by vanishing locus of the spectral determinant. Now without using our conjecture you are not really able to compute that in close form. But because of our conjecture you know that this is equivalent to the vanishing locus of this function here. And this you can really compute exactly in close form because you can compute thanks to the connection with topological string. And in this particular case actually this is given by a theta function and the zero of this function are given by this expression here. So e to the energy are the eigenvalue of this operator and this is what determine the energy level. So n equals zero if you solve this equation for the energy give you the first energy level and equal to one, the second and so on. And usually it is very rare to have this kind of close form expression for the quantization condition also in standard quantum mechanics. Like you have the harmonic oscillator you can do this but you don't have a lot of other example in which you can do something like this. But now with this construction we have all these operators that come by quantizing the mirror curve for which you can do this kind of things. So in that sense we have a new family of exactly solvable problem in spectral theory. And then the second statement is about the canonical partition function and this is something that Marcos also discussed this morning so I will just review that quickly. So the partition function of topological string is given by this expression so this is the grand potential that I was introducing before. This is like the airy contour and the conjecture state that this is equal to the canonical partition function of the gas. So in that sense we give a new meaning to the partition function of topological string from the point of view of operator theory. And this is basically just a combination of spectral traces. And in that in this way as Marcos was saying this morning you have a non perturbative definition of some topological string on toric background in term of operator theory or matrix model because this you can write explicitly in term of matrix model. Now also an important point of this construction and this conjecture is that actually is testable. So there are by now several tests and also application and generalization which have been done and all the time this test and application and so on they all strongly support this conjecture. But we still don't have a proof of the conjecture and in the rest of the talk I will focus on a certain limited case of this duality that things simplify in such a way that actually we can prove that one. So the limiting case that I would like to study is the one that you obtain by doing the geometric engineering. So it is known since a while now that if you take topological string on certain type of manifold like local p1 times p1 and then you take a certain geometric limit you end up with supersymmetric gauge theory in 4 dimension. Now in the particular case of local p1 times p1 what you do when you take this limit is that like you blow one of the of the cp1 and you shrink the other and you do this in a precise way. And then you obtain a cyberwitten theory in SU2 cyberwitten theory in 4 dimension. Now this work for topological string but also work for what is called a refinement of topological string. Again I am not going to go in the detail of this refinement the only thing is that I want to say about it is that basically you can deform topological string by adding an additional coupling so you end up with a theory which has two couplings that typically are called epsilon1 and epsilon2 and in some limit in particular when you put these two to be equal you recover the standard topological string. And in this geometric and general limit this give cyberwitten theory in what is called a 4 dimensional omega background which is characterized by two parameter which are like twisting parameter of the background. And in that limit the full partition function of topological string and the refined version of topological string give you the necklace of partition function in 4 dimension which compute the partition function of this theory here. So this is a well known object which has been computed in detail. Now I would like to see what happened to this topological string spectral to your duality when you take this particular limit. Now if you take the limit in the standard way so you scale this parameter as is typically done in the geometric and general limit you recover another well known conjecture which is the necklace of Schattashvili conjecture in 4 dimension. So this conjecture tell you that the spectrum of this operator this operator is what is called the Hamiltonian of quantum system which is called quantum toda. So the conjecture tell you that the spectrum of this operator is computed by using a cyberwitten theory in a background where one of the two parameter is set to zero. And if you do the standard story this is what you end up with. And you can like slightly rescale the parameter which enter in this geometric engineering and I will explain in a moment what I mean by this. And if you do this then you end up with a new formulation of cyberwitten theory in term of an ideal fermigas. And in this limit basically you can show that the operator that appear is this one and you can show that the spectrum of this operator is computed by using a class of partition function in a particular omega background which is called self dual omega background where these two parameter are equal. And what is nice as I will explain later is not the first time that this operator appear and it appear before for instance is the study of using model and self avoiding polymer but I will get to that later. And also what is important is that in this in this study here we have a relation with penalty equation which allowed us to prove the conjecture so we can prove that the spectrum of this operator is computed by in a class of partition function. Ok, now if we think in term of topological string the reason of having two different limit is related to the fact that the grand potential as I was mentioning at the beginning is composed by two pieces. Ok. So if you take the standard, geometric and general limit what you do is that you send one of the parameter to infinity the other to zero and h bar which is the coupling also to zero and in this if you take the topological string spectral duality and you do this limit then you end up with the necklace of Schotterschwilly conjecture. But then what you can also do is like rescale a bit the parameter and take the basis and the fiber over h bar to one infinity one zero and h bar to infinity. So in this limit this still goes to infinity but this is like kept fixed. And then if you apply this limit to the duality you end up with this Fermi-Gas formulation and you can relate to penalty equation. Like to explain these two different limit from a five dimensional point of view but actually you can also understand these two limits directly from a four dimensional point of view because somehow you can understand them as implementing a different quantization scheme in four dimension. So let's suppose that you have this amiltonian classical amiltonian. So this is the kinetic term and then you have a potential. Then like the standard approach to quantization in some sense what you do is you take x and p to be some operator and this to be the quantum amiltonian. But there is also another approach that people have been used and is what is called the Wigner approach to quantization. So in that sense this is the classical amiltonian and the quantum amiltonian is defined by defining the density matrix in this way here. So this is an operator and again an operator which is like the potential. And then if you do this the amiltonian that you obtain is basically this amiltonian here and on the top of that you have some H bar correction. If we now think at our two limit we can start with the cyberwitten curve which is this curve here. And then basically this you can think of this as the classical amiltonian of the quantum toda which is the system that underlines the necros of Schotterschwilly conjecture. And then you apply the standard approach to quantization and you end up with this operator the amiltonian of quantum toda that appear in the standard necros of Schotterschwilly conjecture. But then you can also do a change of variable and this is what you obtain. And this actually as we will see later is the classical amiltonian that underlines which describe the Fermigas formulation of cyberwitten theory. And then if you take this and you apply the Wigner approach to quantization which is the second approach as mentioned before then you obtain the density matrix for the cyberwitten gas which is the operator that was appearing in the other limit. Ok, so this is just to motivate that from a pure four-dimensional point of view without using the limit. Ok, now I would like to see more concretely what are the quantity that appear when you take this limit. So we start with the topological string side which goes down to the cyberwitten side by using this geometric engineering. So in at the level of topological string this is the object that we have and then when we implement this limit actually we end up with this object here which is a known object that is called the necros of Okunkov partition function in four dimension. So this is defined from the necros of partition function which is the one that compute the partition function of cyberwitten theory in this case in what is called the self dual omega background and this depend on two parameter one which is more like a geometrical parameter this kappa which is related to A A is the cyberwitten period of the curve and then you have the other and another parameter this t which is related to the instant uncounting parameter which is like the gauge coupling and this epsilon here is the parameter which characterize the omega background because we are in a situation where the two epsilon are equal and then the necros of Okunkov partition function is obtained by shifting this parameter and by summing over old shift so this is an object which was already known and when you take this particular limit you recover this object here. Everything I am going to discuss is this rescale limit. If you compute the standard limit you end up with the quantum toda in this part here. So what I am going to discuss is this limit all the time which is not that different. I mean then is geometric engineering we have to small rescale a bit parameter because here the parameter that appear here are rescale. So this is in one side we obtain this necros of a Kunkov partition function for SU2 gauge theory and if instead if instead we go on the other side after we perform this four dimensional limit we obtain another Fermigas which this time is described by this density matrix here. So this is like a non-standard kinetic term and this is like the external potential. So we have again an ideal Fermigas in an external potential. This is again a trace class operator, it has a discrete spectrum so everything is well defined. Now if we want maybe get like a feeling of what this gas is like we can look at the classical Hamiltonian so this is like the kinetic term and then you have the external potential and at large energy this is what you obtain so large energy mean large p and large x so we just have like a relatively simple ultra relativistic gas in an external potential Now if you are familiar with ABGM theory, actually this gas is very similar to the gas of ABGM theory basically the only difference in ABGM is that instead of kosh you have log kosh and this parallelism actually was the reason why sometime ago together with Marcos we studied this kind of operator but what we didn't know at that time is that this give you a description of cyber width and theory and this is basically what the topological string spectral duality tell you in 4 dimension so it tell you that the cyber width and theory in this self dual omega background admit a dual description as an ideal fermi gas and in particular the necklace of a kunkov partition function is identify with the spectral determinant of this gas so again here we have like kappa is a geometrical parameter in this side instead is like the chemical potential of the gas and t instead is related to the instanton counting parameter or the gauge coupling and here instead give you the strength of the external potential now more precisely as I was mentioning this tell you that the spectrum of this operator is computed by the necklace of a kunkov partition function in this particular background and actually we can prove it and we can prove it because we can show that this object and this object they both satisfy the same panlevé equation which is a panlevé 3 equation and they have the same asymptotic so in this side you have to use some relatively recent result as I was showing you are at the beginning and in this side instead to show that this satisfies this panlevé equation there are some slightly more older results so more specifically the necklace of a kunkov partition function or the spectral determinant compute what is called the tau function panlevé 3 equation that Misha was mentioning just before so here there is a plot of this so there are two parameters the T which is like the instant on counting in gauge theory correspond to the time of this equation and kappa instead is related to the initial condition so this is tau function of the panlevé 3 equation and the zero of this tau function give you the spectrum of this operator here and this equality as I said before is related to the A period so kappa is this so is A over epsilon both parameters are scale now so far we are focused on this necklace of a kunkov partition function which correspond to the canonical partition function of the gas you have this spectral determinant now what I would like to do next is to look at the canonical partition function which correspond to the necklace of partition function directly so if we take this Fermi-Gas point of view once you have the density matrix you compute the canonical partition function in this way here so you have to sum over all permutation and then you can use identity which is called the Cauchy identity to rewrite this sum of the permutation in term of a matrix model and this is what you obtain so this is a particular case of a no 2 matrix model which were studied by Kostov and also other people and is characterized by this interaction term here and in this particular case this is the potential that characterizes auto matrix model and this is a relatively known matrix model because it appears before in the literature for instance in the study of 2 dimensional easing model and also in the study of non-contractable polymer on a cylinder so in that sense with this 4 dimensional duality we give another meaning to this matrix model which is the fact that it computes the necros of partition function because on the cyber written side the canonical partition function actually corresponds to the canonical partition function in what is called the magnetic frame and what this duality is telling you is that this matrix model is a compact way to write this canonical partition function oops so in particular for instance the perturbative cyber with an expansion of the necros of partition function coincide with the toughed expansion of the matrix model so we have a matrix model here it depends on 2 parameter n and t and the toughed expansion of the matrix model can by taking n and t large but at the same time this ratio n over t is fixed and if you study the matrix model in this regime you can show that it has this kind of expansion here where these are called the genus g free energy of the matrix model and there are some techniques directly in matrix model which allow you to compute this perturbative expansion this genus g free energy and this was done by some people and then on the other side we have the perturbative expansion of cyber with an so you can show that the necros of partition function has a perturbative expansion of this type here where this quantity here which are called the genus expansion can be computed in a relatively fast way by using the holomorphic equation and you can get you can compute this up to very high genus in a relatively small amount of time and then what we found basically with this construction is that these two coincide and the dual period of the cyber with n theory is identified with the toughed coupling in the matrix model now there is one more comment that I would like to make and it's okay so this this matrix model here you can evaluate this matrix exactly matrix model so if you take n equal 1 for instance this is just like a simple relatively simple integral and is given in term of a best self function so when n is equal 1 this is not very complicated but actually you can also get an expression for higher value of n and this is because there is an underlying tba system of equation which allowed you to compute in principle this matrix model exactly n by n and this means that actually it's possible to evaluate the full necklace of partition function in the magnetic frame exactly by using this tba technique instanton expansion and now I would like to discuss a little bit more this relation with panlevé equation so in this talk basically I focus all the time on this local p1 times p1 geometry and when we take this limit we recover the panlevatory equation but actually you can do the same also for other toric calabiao and in that case you recover a different type of panlevé equation so on that side basically by computing this limit you can reproduce the result by many people here and in this side instead you can basically provide a solution to this panlevé equation here in term of an spectral determinant or a matrix model and this you can do it very explicitly now in this particular case this solution is the one that was proposed in the logical but in principle you can do it also for the other panlevé equation here now just to summarize and conclude so we have a new relation we have presented a new an exact testable duality which relate topological string or enumerative geometry and spectral theory operator theory or matrix model and then I show you that there are two ways in which we can compute this four dimensional limit in one way you recover the standard in the standard limit in some sense you recover the Necrozov-Shatashvili conjection in four dimension in the other limit instead you obtain a new formulation of cyber written theory in term of an ideal permigas and you make contact with panlevé equation so this allowed you to give a proof of this conjection in this particular limit and also it give a new meaning to this relatively famous matrix model which was describing eting and polymers model and this give you the partition function the Necrozov partition function in the magnetic frame which we can write in a compact way as a matrix model now here are some open problems that may be interesting to look at so the first one is now we have really a lot of test of this duality but we still don't have a proof of this so it would be nice if somehow we can translate this proof that we have in four dimension to the full five dimensional case but this is not clear whether you can do it or not but ok and another point is that the topological string spectral theory duality here was generalized to the open sector of topological string so it may be interesting to study the 4D counterpart of this generalization to see what you obtain but so far we always focus on SU2 gauge theory where we have basically the solution of Li-Sovian collaborator and it would be interesting to see what happen when you take this limit of the SU1 geometry in general for any bigger than 2