 OK. Zdaj sem zelo bošnji organizator za ovaj opartonit. OK. So, in tudi, sem bošnji bošnji bošnji bošnji vse formulaciju cyber-witten teori. Vse bošnji bošnji bošnji? OK. Zdaj sem bošnji bošnji vse formulaciju cyber-witten teori korobov drowning vzselur z prvičči takov Xiao tbookihform. Z zelo bošnji bošnji bošnji bošnji bošnji za karabiraciju vse vse katnice bošnji svojoulsion.. Captain in vznik. Now, from a more generic point of view, in this talk we are interested in dualities, and in particular in dualities which relate on one side, something like geometry, string theory and gravity, and on the other side we have operator theory, matrix model or quantum mechanics. We have two quite different theory, kaj je vzljavil na zelo tudi, kaj je vzljavila na zelo tudi. Vzlušaj, ne bomo vzljavili po vzlušenju, pri vzlušenju zelo, ki je vzljavila, ali sem nekaj delošnji vzljavil na njim zelo. In nekaj vzljavil, da se početite, je vzljavila na konjektu. Zelo tudi konjektu vzljava na nekaj zelo, in da sem čudovila, ki je vzelo in seksijo teori in modulji spas. In na zelo je minimalne model, ki je vzelo in teori. In vzelo je, da je več vzelo in teori, ki je vzelo in teori, in da je vzelo in teori občaske vzelo v panleve, ki je, da je, ki je, ki je v panleve, vzelo v panleve 1. To je počkaj izgled, in ne zelo vzelo na to, da je izgled. Vzelo, da bo vzelo, da bo vzelo, da bo vzelo, da bo vzelo, da bo vzelo, da bo vzelo, da bo vzelo, da bo vzelo. Prepoč, da je izgled in konkretne izgled, Mičil se ga je nekonplikata tjera, ki ga čakti teori, vsak facilities teori, kaj discussa taj medokratificne teori. V zelo vzelo, da je kaj čast, da se zelo na dolječi, da je doberoč corrected v sej teori somšel, očasjivo v sej teori, v sej teori, v zelo v osloženju, in očasjivo v sej teori, na zelo, da vse predrožite drugi dodal in drugi dobro. Trenut, da mu to jaz se bila način mač, in zdaj za zelo bilo tamo pej vse izstand i iz mam. Kaj je dobro tik pat bonuses, ovo je čas, ko je dobro dobro Petrojta, skupaj T. In je to, ko Markoš je dobro vse postočil. pri svoj tem, z sedim z vršenju, v njo najtačno je topologična string teori, v tebi manjfoz, ki je tori, Kala Biao, in svečna je matematika, je taj enumerativ kteori. Na vzelo, nekaj je kvantomekaniks, in včetno, in ideal, fermigaz. Toto je vse treba operatora jev. So, so.. Stravna spke connects 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 작은Нsigu along this theory, in four dimension. And on the other side, you have again and another idea Fermigas in začem izgleda je jeznačno potencija. A ne bo bilo vse na tega lepša, da počeš različimo tega zelo, ker tega najbolj občinja je ta teori in ta teori, da se tudi vse nekaj zelo, načo je nekaj vse. Tukaj, da se vzelo, lepši bi smo pričo vse, tudi pričo občinja v lisovi, in pričo vse, a Misha, ki sem je pridem, in da bomo občastil, da se so umilili v nekaj neko velikoskov, ki se zamologi, ki z tem nekaj nekaj nekaj nekaj nekaj. OK, zato sem pridem, da se je počutilo nekaj neko, izgledam, da ne začnemo, da smo tko teori, v teoriji teori, T Here is a theory which studies on maps that goes from a Riemann surface of genealogy into a certain target space. Now, in this talk, I will always focus on the case in which the target space is a kalabbeya manifold, which is called a local P1 times C1. So this is constructed by taking two copies of P1 and then you put a binder. Fizikali tudi mapi prišličiti trajektori v terzovstvih, ki se vzvegačimo in v zelo. Na zelo, da Markos je tudi radila, je to prišličite v nekaj, zato ginovz-gifrij energiji, ko je tudi tudi tukaj. Tjega je tukaj geometričnog parametra, kaj je vzvegači doberi tudi zelo. In tudi genus G-free energičnega vsega je zelo vsega tudi mjelj. In tudi, da se zelo vsega, je, da se organiza genus G-free energičnega v komandu generacijnega, ko je zelo vsega Gopakumavafa-free energičnega in je to obzir, ki je tudi. So G S is what is called a string coupling. It gives a strength of the interaction of the string in space-time. In of the sum, these here are numbers, they are called a gopacumava fin variant. And they are related to the counting of these maps. And then you have these p's here, and here you have E to the keller parameter. Nowa bark ne arrived also mentioned yesterday. If you take for instance g equal to zero in you have one over sine squared. So every time this is like a multiple of pi? You'll get up all. And here, like you do, you have a summation, so you have a dense set of pole along the real axis. So this function is somehow problematic kako se počutiti na vse rejalaks. Vse ga je tudi, da se vse zelo počutite z nekaj nekaj, da vse zelo počutite zelo počutite, da je to, da je vse, zelo počutite, ki je zelo počutite, zelo počutite, a potem vse zelo počutite. Vse zelo počutite je vse zelo počutite, ki se počit počit, zelo se je bila zelo vsehoj velikom vzpejno, zelo je dobro, zelo je je vzpejno vzpejno, zelo je je zelo vzpejno vzpejno, zelo je nekazov, vzpejno vzpejno, nekazov je našel, kako je našel, s ko počit, kako je vzpejno vzpejno, je to izgleda, in je začo, in varjanja, ki karakterizuje vseh vseh geometriji. Nisem tukaj, da imamo vši detail, da ne vidimo. Znamenj, da imamo nekaj kvaliti, kaj je potenšel. Tukaj je vse vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh. Vseh, ali je viča in objeht, ki imamo Ankanonical Partition function. Tukaj je vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh vseh. Tukaj je zelo vseh vseh. Njah sem vseh in vseh. Vseh toga proch in voči generationsu. kaj je izgledal, očeščenho se swoje drugi tudi, je očeščenho se srednje psijeljna teorija. Naočeščenho se whistlej je visiturbel in napomisla, ki je dobro, občaj, ki se je odstavilo, kaj termineč je nekaj kaz, ki je kajega zenojouchaja na stafično izprav. Ako ta skočnja je dobro, nekaj zelo pri stafičju odstavljala odstavljala, da bo se v tende, kako je dobro v teo, pri trvi neko teo, nekaj je dobar, is the one whose density matrix is constructed by quantizing the mirror curve, as Marcus was mentioning this morning. So in the case of this geometry, the local P1 times P1 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. So basically this is the density which describes the Fermigas, which is dual to topological sting on local P1 times P1 according to our duality. So this is a discrete spectrum, the row is a trace class operator and it is 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 Fermigas in statistical mechanics. So it's the determinant of 1 plus kappa rho, kappa play 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 common 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 vanish 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 denomerative 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. And so this duality can be summarized in two precise statements. 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 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 you have 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 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 gives you an exact relation between enumerative geometry on this side because this computes the enumerative invariant of the background geometry and a spectral theory because basically this is what contains the spectral information of the operator here. So in that sense this duality gives you a new family of exactly solvable problems in spectral theory. So a new family of operator for which you can write down an exact closest version for the quantization condition which on 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 very 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 in exactly an in close form because this 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 determines the energy level. So n equals zero if you solve this equation for the energy give you the first energy level and equal one, the second and so on. And usually it's very rare to have this kind of close form expression for the quantization condition also in standard quantum mechanics. You have the harmonic oscillator but you don't have a lot of other examples 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 spectra 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 function of topological string from the point of view of operator theory and this is that these basically are just a combination of spectral traces. And 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. You have seen some tests this morning 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 limiting case of this duality in which things simplify in such a way that actually we can prove that duality. So the limiting case that I would like to study is the one that you obtain by doing the geometric engineering. So it's 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 four dimension. Now in the particular case of local P1 times P1 what you do when you take this limit is that you blow one of the Cp1 and you shrink the other and you do this in a precise way. And then you obtain cyberwitten theory in SU2 cyberwitten theory in four dimension. Now this work for topological string but also work for what is called a refinement of topological string. Okay, again, I'm 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 the four dimensional omega background which is characterized by these 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 four dimension which compute the partition function of this theory here. So this is a well known object which has been computed in detail. Okay, now I would like to see what happen to this topological string spectral duality when you take this particular limit. Now, if you take the limit in the standard way so you scale this parameter as it's typically done in the geometric and general limit you recover on another well known conjecture which is the necklace of Shatashvili conjecture in four dimension, okay. So this conjecture tell you that the spectrum of this operator this operator is what is called the Hamiltonian of quantum of a 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. Okay. And okay, so if you do the standard story this is what you end up with. Now 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 actually it's not the first time that this operator appear and it appear before, for instance, is the study of easing model and self avoiding polymer. But okay, I will get to that later. And also what is important is that 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. 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. So if you take the standard geometric engineering 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 Schottaschvili conjecture. But then what you can also do is 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 so 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 a penalty equation. So this is like to explain these two different limit from a five dimensional point of view. But actually you can also understand this to limit directly from a four dimensional point of view because somehow you can understand them as implementing a different quantization scheme in a four dimensional. So let's suppose that you have this amiltonian this 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, 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. So if we now think at our two limit we can start with like the cyberwitten curve which is this curve here. And then basically this you can think of this as the classical amiltonian 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 describes 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 that I was mentioning before then you obtain the density matrix for the cyberwitten gas which is the operator that was appearing in the other limit. 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 quantities 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 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 Necrozov Okunkov partition function in four dimension. So this is defined from the Necrozov partition function which is the one that computes the partition function of cyberwitten theory in this case in what is called the self dual omega background and this depends on two parameters 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 on counting parameter which is like the gauge coupling and this epsilon here is the parameter which characterizes the omega background because we are in a situation where the two epsilon are equal and then the Necrozov Okunkov partition function is obtained by shifting this parameter and by summing over all shift So this is an object which was already known and when you take this particular limit you recover this object here Now everything I am going to discuss is this rescale limit If you compute the standard limit you end up with toda, quantum toda and this part here So what I am going to discuss is this limit all the time which is not that different I mean aden is geometric engineering but it's just that you 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 Necrozov Okunkov 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 to 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 means 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 some time ago together with Marcos we studied this kind of operator but what we didn't know at that time is that actually this gave you a description of cyberwitten theory And this is basically what the topological string spectator duality tell you in 4 dimension that the cyberwitten theory in this self dual omega diagram admit a dual description as an ideal Fermi gas And in particular the Necklas of the Kunkow partition function is identified 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 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 Necklas of Kunkow 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 other results More specifically the Necklas of Kunkow partition function or the spectral determinant compute what is called the tau function of a 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 instanton counting in gauge theory correspond to the time of this equation and kappa instead is related to the initial condition So this is the 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 before is related to the A period So kappa is this so is A over epsilon both parameter are scale Now so far we are focused on this Necklas of Kunkow 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 Necklas of partition function So if we take this Fermigas point of view once you have the density matrix you compute the canonical partition function in this way 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 an O2 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 this O2 matrix model and this is a like relatively known matrix model because it appears before in the literature in the study of two-dimensional easing model and also in the study of non-contractable polymer on a cylinder So in that sense with this four-dimensional duality we give another meaning to this matrix model which is the fact that it computes the Necklas of partition function because on the cyber-width side the canonical partition function 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 Ops So in particular for instance the perturbative cyber with an expansion of the Necklas of partition function coincide with the tough expansion of the matrix model So we have a matrix model here it depends on two parameter n and t and the tough expansion of the matrix model is defined 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 and 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 this is relatively hard but this was done by some people and then on the other side we have the perturbative expansion of cyber-width so you can show that the Necklas of partition function has a kind 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 anomaly equation and you can get like you can compute this up to like very high genus in a relatively small amount of time and then what we found basically with this construction coincide and the dual period of the cyber-width and theory is identified with the tough coupling in the matrix model now there is one more comment that I would like to make and it's ok 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 simpler integral and it's given in term of a Bessel function so when n is equal 1 this is not very complicated but actually you can also get exact expression for higher value of n and this is because there is an underlying tba system of equation which allows you to compute in principle this matrix model exactly n by n ok so in partic 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 and not just as an instanton expansion and ok now I would like to discuss a little bit more this relation with panleve equation so in this talk basically I focus all the time in geometry and when we take this limit we recover the panleve equation ok but actually you can do the same also for other kala biao and in that case you recover a different type of panleve 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 panleve equation here in term of spectral determinant or a matrix model and this you can do it very explicitly now in this in this particular case this solution is the one that was proposed by its homological but in principle you can do it also for the other panleve equation ok 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 in the other side 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 recover the nekrosov-šatashvili conjecture 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 panleve equation so this allowed you to give a proof of this conjecture in this particular limit and also it give a new meaning to this relatively famous matrix model which was having heating and polymers model and this give you the partition function the nekrosov partition function in the magnetic frame which we can write in a compact way as a matrix model now here are some open problems ok that may be interesting to look at so the first one is ok now we have like 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 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 generalize to the open sector of the topological string so it may be interesting to study the 4D counterpart of this generalization to see what you obtain and another point is that so far we always focus on SU2 gauge theory where we have basically the solution of please sovian collaborator and it would be interesting to see what happen when you take the SU1 geometry in zoner for any bigger than 2