Simple Gauge-String Dualities

Please take a seat. Uh all right. So we will kick off the day with the fourth lecture of Rajes Gopakumar on string gauge dualities. >> Thank you. Uh thank you Franchesco. Um so uh we yeah there are some right so um so this is the final lecture in which I'll introduce as promised the bodel version of uh uh these uh simple gate string dualities. Uh so uh so yesterday we had I I talked about a model duality uh in which I motivated the duality in uh uh couple of ways. Firstly from the matrix model side how we had a combinatorial way which uh in terms of permutations which is equivalent to counting branch coverings. Uh and how using the um the prescription of this trouble reconstruction of closed string world sheets from fineman diagrams. uh we can uh we can kind of guess what the a model what the duel is which uh was and and then I proposed a concrete a model duel and we could actually I didn't have the time yesterday to show but uh uh through this connection to the sequels to one string theory one can actually check that these correlators uh agree of the a model and uh what we are currently what I couldn't present was how to show from the world sheet point of view that uh the so we know that the answers for the a model proposal agree with that of the matrix model they reproduce them at all genus but what we would further like to go uh forward and show that indeed the a model description that I mentioned gives you exactly the kind of picture that the matrix model gives. Namely, in terms of these uh counting these belly maps, these special maps from these particular points on the modelized space, this uh localization on these uh special arithmetic points uh which are given by integer treble lengths. So that's a stronger statement which we would like to show which I believe is true. uh uh but the equality of the A model with the matrix model is something that can be checked. I just wanted to clarify that that that part the as far as the duality is concerned the uh the equality of these two string theories. we know that is true and we want to make a kind of a more refined or toological kind of a check of that duality which uh hopefully will also be true but today what I will show is more another world sheet description so it's almost like embarrassing to have multiple g string dual descriptions but this will be will look very different though in some sense morally it is probably the mirror to the description that I um described yesterday though that is still not completely clear to us how exactly it is uh the mirror but uh I mean it's plausible uh this is work with Aleandro Guaketto and Edward here which is really hot off the press in some sense meaning it's not yet off the press it will be hopefully appearing in in the next couple of days and here the world sheet picture we have a very complete control over and we'll be able to uh fully address the challenge that I uh outlined in the very first lecture uh um which was uh so basically do we have a dictionary um u an operator dictionary which sort of uh uh uh relates a gauge theory single string gauge theory observable. single trace gauge theory observable to a single particle string vertex operator such that uh such that and the uh part two in some sense is to show that there's correlator in at any genus uh and with these couplings uh is equal to the integral of again a string correlator with the TIS these to couplings becoming now sort of part of the string background. in some sense resuming the whole perturbation expansion. Uh so that uh um that's uh uh so that's what we will uh be able to address this and when I say this uh we want an autonomously defined uh string theory that will reproduce these uh correlators that will uh uh for whose rules will uh reproduce the right hand side and u uh uh and u and have a effective computational recipe which will uh which will uh lead uh lead to a so which we can check uh indiv independently and which for which in some sense we would ideally like to even sort of tautologize and we'll see how this is done in this uh uh this thing and uh in particular uh I as concrete challenges in the first lecture had mentioned uh a few correlators like the threepoint function of uh uh of these traces but in a in a quartic potential for instance uh and I'd mentioned the genus one uh onepoint function for instance so we'll see what uh explicitly so towards the by the end of this lecture I'll get to the point where we can set up this uh calculation and then Edward will show uh he and Alessandro have a very sort of now created a sort of a assembly line for sort of generating all these correlators and he'll show you some of the I think impressive uh checks that you can make. So anyway, so that's the goal for today um to answer these questions in the context of this B model. By the way, I just want to say that here also. So we are not doing this here in the uh in the directly in the approach that we uh that I outlined yesterday morning namely through the uh resuming the fineman graphs. will sort of directly have the string background uh which will encode these couplings uh but um uh but I believe there is should be a way to also uh view this in terms of the fineman diagram picture uh perhaps from the sort of the f-type u picture rather than the vtype picture that we had uh fineman diagrams that were uh reconstructing the A model. uh but that's for the future. So um so let me state the uh the proposal and uh unpack it a bit. Uh so we'll uh so the B model will be a B twisted. So uh so the uh these topological string theories come with two inequivalent twists. The A model is the sort of the A twist and the B model is the sort of the uh the other inequivalent twist that you can do. Uh the B model twist. uh so I won't have time to explain these uh but in the simple cases where there are conformal field theories it's essentially a question of uh whether you twist the stress tensor in the same way or in the opposite way on the left and right uh but you can look up some of the uh very nice lecture notes including by Marcos on topological strings for more uh uh details on that. Uh so the B we'll talk about a B twisted so-called Landau Ginsburg theory. Uh so that's a basically a scalar theory. Uh well we'll talk about an N equals to two uh Landow Ginsburg theory n equals to two super symmetric Landow Ginsburg theory that will be twisted. Uh so it uh uh will uh be a scalar field with some sort of landsburg potential or super potential and u and firmionic partners and u this coupled uh uh to 2D topological gravity. So which is essentially the the piece which involves the ghost system of the 2D gravity theory because 2D gravity is not really a propagating theory. Uh so uh so what is the data for this world sheet theory? So the that has to encode the information in the matrix model. So um so this uh the uh the data of the matrix model uh leads enters the string background that determines the string background. And how does that do that? So one way to encode the data of the matrix model is through what is called the spectral curve. So uh this is a remon surface that you can associate uh with uh a general one matrix model like this u and it arises from a simple Schwinger Dyson equations. So stringer Dyson equations are ones where you basically derive a non-trivial identity amongst um correlation functions of field theory or a in this case a matrix integral by using sort of a total derivative uh uh term. Uh so so you um uh take the total derivative of this is equal to zero. It gives you uh um some equation for so this x is just a parameter. Uh it's basically this is essentially capturing the resolment of the u matrix model. So this leads to a curve which I'll write it as uh um uh as is given in terms of the potential and this y is something like minus v prime of x by 2 uh plus the resolvent uh at genus zero. Uh so uh so essentially you compute this resolvent and this uh and the potential is given. This is also given in terms of the potential. It's uh uh easy enough to write that it's a polinomial for a polinomial potential. it's a polinomial and uh so this um uh uh this this definition of y obeys this relation uh that's essentially the content of the uh swinger Dyson equation and uh so it gives you a reman surface if p is a polomial uh if v is a polinomial uh so this is what is known as the spectral curve It can be so this curve can be uniformized uh by uh uh by a coordinate z uh in terms of which you can express both x and y. So we'll now uh we'll look at the uh phase of the matrix model which in which it is a single cut though it can be generalized to the more general case as well and this will change in that case but in the case where it's a single cut we will u parameterize this x uh in terms of this uniformizing coordinate as follows where gamma and delta are basically so gamma is basically the width of the igen value distribution uh in the planer limit um and delta is some kind of an offset so for an even potential which is symmetric delta is zero but more generally it's some kind of an offset so that's so these these do carry information about the toft couplings but in some so it's just a very specific combinations of the to coupling so gamma is some gamma of the tis and delta is also so this uh is information about the there's some information but very kind of coarse information about the uh so it's basically from the if you wish you can write it in terms of the a plus and a minus the end points of the igen value distribution so the igen values might uh in this one cut phase might take some form like this and uh this are essentially related to that uh and um the the y also uh is the one which actually carries more information about the potential and it is uh is given in so these u cases are determined by the tks. So the potential basically uh there's a simple sort of a uh transform of the potential that uh uh allows you to read off the UK case and uh uh so y of zed is basically a again a polinomial in uh these uh uh uh in zed uh so this is a uniformizing coordinate and that uh is one way to uh parameterize this uh just basically parameterize this uh curve and um so um so this these data of the matrix model which are entered in this functions y of zed and x of zed will go into the uh closed string background so that's what I was saying over here the string background will be determined by this data and how is that done so as I said the the theory is a Lando Ginsburg theory. So a Landow Ginsburg theory takes the following form. Uh so it's a two-dimensional uh theory with a super potential uh W. uh so that's so it's basically a scalar theory with a potential or in parameterized in terms of a super potential as I said it's an n equals to two theory so it has some firmians row and s uh I think uh uh I think row del r row bar del r row del bar sibar and row bar del s plus uh So row and si are firmians but it's a twisted theory. It's an n equals to2 theory in which you've twisted. So row is actually a one form firmian. Uh so when you twist it one of the the twist depends on the arch charge and u so one of the firmians becomes spin uh one form uh um spin one whereas the other one is a scalar zero form uh and those are these rows and size and phi is a uh is a scalar field. So this is the this is a two-dimensional field theory and in general it'll be a massive uh theory. So the data here uh in this field theory uh is encoded in uh in this uh is coming from basically these functions. So x this function x which is basically this. So it's basically gamma into 5. So, so we take the super potential to be this uh um and we take the um there's also a a holorphic one form uh that we we will need uh to uh make the appropriate sort of to uh to to when we evaluate the correlators to uh we'll need a holorphic one form which will be essentially also part of the background uh and this is just given by dy fi so this function so I'm just replacing the argument z by five uh so uh so these are uh given by uh this so this data translates into the data of the uh string background that is the proposal and uh so this is uh a TFT a topological field theory. So this any twist to Lando Ginsburg theory is a TFT and we can say many things about it. It was first stud G55 bar. >> Yeah actually things will not depend on the metric. So in the in the twisted theory the things are independent of the metric. It's a BRST sort of uh closed um this thing and um in this case anyway it's a kind of flat it's one dimension anyway so the metric is something you can reparameterize away but um uh uh so so this matters uh theory was this topological field theory was studied I buff offer in the '9s just as a field theory and one uh and the effect of the twisting is such that uh the uh uh TFT path integral uh localizes on constant field configurations And so in other words, so the PH of sigma is basically a constant and it's not just any constant in uh I maybe some of you are familiar with the B model on a calabia or something in which case of course the constant would be any uh value on the calabia but here there's a potential and the value of this is basically the point the critical points of the potent of this super potential. So, so it localizes two field configurations where the field phi takes uh the value ph not. And in our particular case here uh you see that uh if you take w of phi to be this then in our case 5² is equal to 1 which implies not is plus or minus one. So if you just differentiate this you just get uh two critical points uh which are at fi not equal to + one or minus one and uh uh and those are the two values uh over here uh okay so and sorry I have another question >> so in order to define I don't know a good string theory shouldn't one have a CT cft rather than a massive theory So in general if you wanted to build a uh so indeed these were originally introduced to uh define the CFTs. So right now I'm just talking about matter just purely at this stage just topological field theory. These were originally introduced to study the uh n equals to 2 minimal model CFTs and the idea was that these massive theories would flow in the infrared uh to CFTs. But the topological field theories for which the Landow Ginsburg models for which that would happen are the ones for which the super potential is actually homogeneous function of the fields. But here we are not taking that. So this is a massive theory. So but you can consider this of course as a topological field theory. Uh now when we talk about the topological string theory we'll be actually coupling uh so the standard way in which you construct topological string theories is to take a CTF twist it and then couple it to gravity. uh but um you can also consider a larger set of topological string theories where you just consider a topological field theory coupled to uh topological gravity 2D topological gravity because in some sense you're already in a critical theory you you don't have so if you're in a sequels to zero effectively a sequels to zero theory and you need the CTF only to kind in in strength theory we need a CFT only to get rid of all the wild degree of freedom and so on so that the gravity doesn't have any propagating degrees of freedom in the full string theory the 2D gravity doesn't have any propagating degrees of freedom but here anyway 2D topological gravity doesn't have any propagating degrees of freedom so as a topological string theory there's a larger space of theories in some sense you can construct which can be CFTs which are deformed uh but which are still topological and coupled to topological gravity 2D topological gra. So this is something I think that physicists have actually not used so much but if you see in in um I think in the mathematical literature I think they consider this larger space of topological string theories or what we would call topological string theories. I think they call them theseological field theories but um but Essentially in some ways this was already implicit in the original relation that Witten and Concevich uh when they studied uh so Witten talked about I mean in the minimal string theories in the double scaling limit people talked about the two comma 2 m +1 minimal model CFTs coupled to gravity and then you can view them also as sort of twisted theories uh coupled to 2D gravity as Witten showed and therefore can compute some topological information. But if you see what for instance Conservich did, he was considering the full theory which in which you can deform away from just the 2, 2 m plus one but consider all the minimal models together and in fact he had if you wish you have you you you can interpolate between all of them. uh so you can consider more general sort of topological string theories uh which you can't which you wouldn't normally consider uh so you can start with CFTs but consider defamations which are topological and couple them to 2D gravity so that's a sort of long answer to the question but um yeah so u so indeed they originated from CFTS but uh for these topological string theories you can consider uh these more general uh class of theories and um um okay so but let me I'm still not yet coupling it to the topological gravity I'm just talking about the field theory uh so in the field theory the physical states so this is a twisted theory so again uh the physical states are in some cucomology of uh um of this theory and they they are essentially the sort of chyal primaries if you wish in the uh in the language of the n equals to two CFTs. These would have been what would have been uh the chyro primaries. uh and they form uh uh so so the uh so the these correspond to the matter so to say more generally the matter primaries in this uh CTF and this is given in this case by just polomials in phi uh modulo this relation uh the that dw is equal to zero so the relation dw= = to 0 is basically phi squar is equal to 1. So it so the only independent generators are 1 and five the identity and phi in this kyal ring. So in this matter sector with this super potential there are only two sort of two matter primaries. uh so that's uh again the that's why these are topological they have very uh limited content in the sort of minimal model CFTs for instance you would have had 5 to the k + 2 as a potential and you would have had a ring with five to one up to 5 to the k so anyhow uh the um so so This is so there are two primaries. There's a useful basis that you can construct of from these uh two primaries uh 1 + minus 5 / 2. In some sense these are nice because you see the uh critical points are at 5 equals to plus or minus one. So these O plus and O minus are kind of localized at one or the other of those two critical points. So uh so this makes it sort of nice if you wish you have this relation sort of an item important relation that O plus minus uh the squares to one so squares to itself. uh so so you can we'll take this basis for convenience as the uh as that for the matter primaries and um so what buffer had shown for instance in the original uh study of these lands work models is that you can compute because of this localization you can compute the correlators the physical correlators are basically those of these O plus and O minus and in this basis they are actually diagonal and if you wish uh so you can consider this Landow Ginsburg theory and you can actually look at it on a genus G surface. So this could be some genus G surface and uh and the answer is quite explicit as he showed it's basically given by the hessen uh at the so the hessen is just the second derivative of the super potential at the critical points that's what w plus minus refers to evaluated at this critical points and uh uh uh and this omega which is this holorphic form also evaluated at the critical points plus or minus again uh um to the power g minus one. So, so, so this is something that's just a very nice sort of uh evaluation of the path integral and the one loop fluctuation around it that uh allows you to uh write down what the matter correlators are. Okay. U so this is sort of the matter content uh of this theory. Uh now you want to couple it to topological gravity and that is very much more subtle uh that leads to much more uh that yeah so um so the 2D gra 2D topological gravity so you couple uh couple to 2D topological gravity and the now the physical vortex operators of this topological string theory. Now so you have the matter sector and the 2D gravity sector and the mat the physical operators are essentially these alphas where alpha is plus or minus. So uh so yeah uh times s to the k where s is basically so s is a two form on the modulized space of uh the reman surfaces which essentially is has a geometric meaning but physically it's some uh it's a kind of what is called the gravit So these are called gravitational descendants or the sort of dressing of the matter with the gravity fields. Basically they are built from the BC ghost and you can find in the old literature explicit forms of this uh but we'll use it in this form because this is the uh this is the form uh literally the form in which you can do integrals over the modulized space with so these will be uh integrals over uh these are forms characteristic classes on um the modalized space of genus G surfaces. So uh so they take this form where K can be anything I mean so uh um so you can have s to various case. So this whole thing is some 2k form uh on modalized space. >> So yeah >> I think there might just be some confusion that it's there's also a sigh on the left hand side for the matter theory. I think people sorry this sigh is not the same as that sigh. Maybe I should put some capital S or something here. Thanks. Yeah, this S is not the same as that S. Good. Uh so um so they so they are of this form this O alpha tensor side to the K. >> Yeah. Yeah. >> Yeah. I mean when you didn't couple it to the top uh grav 2D gravity yeah then probably it was some uh it was some coomology of uh some topological charge that was these kyal primaries right this O plus minus >> I mean these are yeah this is the Q these are the Q exact states the Q closed states >> when you introduce gravity like the definition of that Q will change or >> Yeah Yeah. Yeah. Yeah. Of course. Because you can think you can actually write down a l people have written down a lrangian version of this uh of the 2D gravity twisted 2D gravity in terms of the Louisville field and the super partners. Uh yeah, you can write down and write everything in terms of that in the old papers of Berlin days and so on. In fact in ICTP spring school of 1990 or something like that I think you'll find quite detailed discussions of >> but from probably from that definition it follows that uh this will be like a >> this will be the most general sort of uh so there'll be a Q matter plus a Q gravity >> okay okay u so okay so now kind of we are ready to at least the uh dictionary the first uh this thing. So the trace m to the k uh uh corresponds to this vk which will be some linear combination of these things and what and there's a very precise linear combination. So you sum over these alphas. So you have these O alphas and each O alpha comes with some polomial in this size. So if you take M to the K, it comes with a polinomial of degree K and that polinomial is or degree K minus one actually. um and this polinomial so to the d uh so uh comes there are some very explicit coefficients here which I can just write down to show that there's it's just some combinatorial coefficients uh which which also depend on uh um which uh depend also on this gamma which enters in the super potential gamma gamma and in general for delta also I think this is written for the case when delta is equal to zero uh so uh and you have uh No, no, both are the same alpha. >> Yeah. So, this is just plus minus one. Sorry, it's plus minus one. Yeah. Yeah. This is a D. This is a D. Yes. And that's an alpha. Sorry. Uh so and this is some combinatorial factorial form. Uh so uh so you can write down some dictionary like this and uh and you see that the um uh that even this dictionary knows something about the background but it is encoded purely in this gamma and more generally in delta as well. uh um and also to some extent in omega alpha uh so uh that um is so in the operator dictionary knows something about the background which is uh understandable but the u so that's the uh so that's what this part of the this thing is the more complicated part is to now uh define how what the correlator is uh because it's no longer just a matter kind of a correlator like here for instance this these were simple matter correlators now you have the matter is coupled to 2D gravity and the in the action as well the gravity also couples to the matter so you you have a kind of a combined system which um so the correlators so if I want to so that's what I will um uh use the following kind of a notation for to uh this double brackets to uh to sort of uh now if I want to compute these in the matter topological gravity plus matter at some genus. Uh G this is related to the correlator without the matter. So just of the Landau Ginsburg like this uh but with some operations which I'll uh uh which uh which are quite complicated operations and this was one of the things that uh I think it were in the general formalism ofological field theories that mathematicians and others have developed given Tal Telman and others it was sort of understood to take this form. But I think this was one of the heroic things that Alessandro and uh Edward really worked this out very explicitly so that we now have a uh we have a very precise algorithm on how to convert how to rewrite the uh correlators of the matter plus gravity part into uh into uh into be um uh in terms of this. So this is uh so this is some kind of it's implicit uh so this is defined in a sort of a non-trivial way. Um I'll just maybe draw some pictures to illustrate this and we'll see a little bit more of it when Edward shows uh some of the explicit expressions. But um u but uh the u so let me just uh illustrate what this is. But so there is a definite algorithm which basically takes these correlators and writes them in terms of uh these with some additional side classes. uh so corresponding to the matter part to the gravity part and sort of and then of course the matter this part of the uh path integral you know how to do and then you're left with something that involves just the site classes uh and other characteristic classes which you can then integrate over the modelized space. So there's a very definite algorithm which allows you to to do this. Um and u this also has a uh has a a very natural interpretation uh this these operations which um uh I won't explain but let me just uh draw the sort of contributions. uh so these basically take into account so what let me say what why you have something non-trivial here uh so the point is that uh the 2D topological gravity when you couple to the matter fields then you get uh you have to take into account what the correlators are uh even at um even when the uh remon surface degenerates so from the degenerating reman surfaces from the uh contact terms that involve the 2D gravity fields and the and the matter fields you get all lots of additional contributions. So this is a way to keep track of those additional contributions. So uh that's one way to sort of say why you get uh u sort of additional you why this it's not just some factorized into some matter piece and some uh gravity piece because uh the gravity couples to everything and you get uh both in terms of the degenerating reman surfaces kind of uh contributions contact terms or pinching terms from there as well as from the uh 2D gravity terms themselves when they uh collide with the matter terms. So um so I'll just kind of illustrate this I think uh in just in pictures. So um for uh say a single onepoint function at genus one uh so you get a piece which is sort of just the O alpha. So this is sort of the one the usual one the matter piece at genus one. So you get this is equal to a sum of terms uh which include this plus something that comes sorry then from a pinched taurus like this. So, so this is one degeneration of the Taurus where it pinches and of course you can have this O alpha. So there's a a degeneration contribution which itself can be written in terms of some uh that's where this R comes. Uh uh so this from this degenerations uh um and you can think of it in terms of uh sort of um yeah um so this can be viewed in terms of something where you kind of remove this part, compute the matter correlator on the rest of this and add a contribution from here. What the precise contribution is I'm not going to write but that's what this R essentially captures. So so essentially the matter uh correlator is okay. it gets sort of a uh regular kind of a contribution from this part. But this piece get gives you an additional uh piece that maybe Edward will in fact write down what the classes are corresponding to that. And then similarly there are some in this particular case there are uh um uh some others as well uh which um uh correspond to essentially uh uh okay maybe I won't write down all the different pieces but there's sort of uh you you you can consider the operator or alpha inserted on a disk and uh you have also um uh so you have an you basically you again consider a a disk so this is a contact term coming from the gravity coupled to this uh or alpha so it's really this is the piece which is the tp uh well uh in this case it's the uh so there's a piece corresponding to r there's the piece corresponding to T, which is uh when you have a gravity insertion in the um in uh in any of the other pieces uh in any uh so you get you get a site class from here which will then uh which you then integrate over. So there are uh okay I I think I won't be able to explain it very uh clearly in this limited uh period of time but you get a very precise way in which through these r and t you capture the different degenerations the different ways in which the contact terms of gravity uh couple to these matter fields. Um uh so uh that allows you to uh to write down this complete object in terms of the usual matter correlators plus terms involving uh classes on the modalized space and I think maybe in the example that we'll see it may become a little more clear how you get then something that uh can then be integrated over the model space. So there is a very clear a very definite algorithm by now which enables you to do this. So um I wanted to have um uh Edward show some of these results and then I'll make a few additional comments. Um but let me just make a couple of these comments and then we'll see the see. So I'll again not um describe this it will be there in our paper. Uh the double scaling limit arises very naturally here. You can take the double scaling limit on the uh on the string theory side and on the field theory side. The field theory side you zoom into that critical value of the coupling. You remember for the quartic theory we saw that things blow up when say t4 approaches t critical which is was something like 1 by 12 a and uh uh so you can sort of uh so that essentially amounts to zooming into one of the critical points of this potential let's say phi equals to + one and u and you can show that this operator dictionary everything just simplifies. There's essentially the matter part becomes very trivial. There's essentially only one critical point and uh so the matter correlators are just some numerical factors. uh you don't have this sort of alpha equals to plus minus uh contributions and this piece also simplifies to just become the uh the piece that was identified by concervich as the leading piece that gives rise to the u so in the double scaling limit the matrix model goes to the usual uh usual sort of zooming into the edge of the cut uh uh of the of the cut the and the string theory goes to sort of the pure effectively purely 2D topological gravity uh you the matter part essentially becomes uh trivial and so that's one uh thing so that works out very uniformly. You can see that there's a kind of a uniform way of taking that limit. Um and why does this work at some level the so of course you can check uh and we and Edward will just show you all the checks but ultimately it works because both sides obey something called topological recussion which is a machinery that was developed by Czechov Oranten and so on to as a general way for us to compute all genus correlators using the spectral curve data and some other data u of the matrix model. And that mathematical machinery in in some ways is something that that general formalism of topological recussion is something that these topological string theories orological field theories obey. and uh a and uh with this data and this identification uh you can actually map the two uh uh to each other. So that's ultimately one way of seeing structurally why these uh uh these agree. And uh so the last thing I'll say before I um stop here is about the A model versus the B model. And I think this is something very interesting for the future which is that the fact that there are these two different pictures is very suggestive I think because in this picture that we were describing today these correlators are given by some integrals over some characteristic classes on modiz space and uh you get you reproduce the numbers but uh you reproduce the matrix correlators but the matrix correlators if you consider them as an expansion um around the Gaussian. We saw that those are all integers. We saw that they are just answers to certain counting problems. Whether you think of them as counting graphs or counting permutations or counting branch covers then integers. Uh if you when you expand it around say t equals to zero in a power series expansion uh there's an infinite amount of data for arbitrary correlators, arbitrary genus. But they're all integers. You can do the same thing here. Expand these also explicitly in powers of t. And uh you the claim is that those integrals over modalized space are also integers. The same integers. Uh and that's firstly a very surprising I think from mathematician I think from the mathematicians point of view it's quite surprising that they should give you integers. Uh but secondly even more interesting is that they are not just any integers. These integers have a meaning. They were counting these branch covers for instance uh which as I said are you can view as basically counting special number of points on the modilized space of reman surfaces. these arithmetic points. That's what the belly maps uh count essent I mean counting belly maps is counting certain set of these arithmetic points certain classes of these arithmetic points. So it's as if on the one hand some integral over this m bargn of some class over here some top form is uh which has a lot of data uh is related to this number which can be viewed as a sum over some points belonging to this modiz space. uh sum these integer points if you wish or arithmetic points. So this number is so it's as if this integral gets contribution only it's sort of delta function localized. So there's some way of rewriting these in a way such that these are some critical points of some mors function or something on this modalized space. But it's very interesting that these arithmetic points are picked out as somehow these critical points. So I think that's mathematically a very uh non-trivial fact uh which follows from the fact that there are these two different a model and b model ways of thinking about this uh uh the same matrix integral. So um anyway, I'll now invite uh Edward to uh to sort of show some of the uh the checks, especially the uh especially the correlators. Yeah. Uh okay. >> Yeah. >> Yeah. Okay. Yeah. Maybe I can go ahead. Yeah. So I I mean by looking at the explicit form that you get on the moduliz space using this s classes is it easy to see that like there is something special about those arithmetic points on the model space >> the whole point at least nothing to that I can say uh okay or at least any of us can say I mean Aleandro is a mathematician who has worked with these classes quite a bit for his thesis and everything and uh it it's quite surprised was quite surprising to him. So uh yeah so it's not obvious why these should be related to some >> but somehow the number you produce after carrying out the integral is exactly this sum. Yeah, >> exactly. >> Because there's some because as you will see Edward will generate these integers for you and uh the matrix model tells you that those are essentially like I showed in my lectures counting these curves. So >> and one more thing about this offset. So this offset uh was somehow not important when you when you didn't have the uh 2D gravity that offset was not important >> important meaning it's enters in the dictionary it doesn't enter here that's what you mean >> correlator like uh was independent of the offset >> so yeah from the the matter theory this is just some additive term >> but somehow there you had to set delta equal zero >> there matters >> in when when 2D gravity comes in. Okay. Okay. But at the level of spectral curve, is it like the how >> just the potential >> cut is away from the >> Yeah, it's an offset like I said. Yeah. It's from the origin. So it need not be symmetrical. >> Okay. So typically if you have an odd potential it won't be symmetrical. For an even potential delta is zero. I think Marcus >> maybe we can wait until we Edward might maybe. >> Okay. >> Let me tell you. So what we're gonna do is we're going to >> need a microphone. >> Uh yeah. >> No no you can use this. We go. Okay. So what we're going to do you know people have uh I heard many times that you know you don't understand something until you can code. you know put it on a computer because once it's on a computer you've really understood sort of the details unfortunately I can't code okay but uh that means maybe I don't understand many things but we have this uh remarkable mathematician who's also a very good coder okay and what he's done is he's sort of taken everything you see on this board and he's sort of put it into a computer code and what it's going to do is that it's going to compute this integrant on moduli space now this is a very very non-trivial thing okay what you should think of doing is you're trying to compute some string theory path integral and you've done all the possible integrals. Okay, something like in the bzonic string. This is saying you've computed correlation functions of 26 scalar fields as a function of the moduli of the surface. Okay, and so what we're doing is this last remaining integral as an integral over the moduli of the surface. Okay, so something like maybe seen in a Taurus. This would be like toao and toao bar of the Taurus. Okay. So instead of trying to compute this as a function of the moduli which usually is a totally impossible task. Okay. So already a twopoint function on genus one is a very difficult thing to do. We're going to what we're going to do is to use this mathematical language that uh uh rees and we're going to spit out directly some differential forms on moduli space. Okay. So like on a d-dimensional space you can integrate a d form. Okay. So what we're going to do is we're going to spit out some forms but in particular there's something very special. So these matrix integrals they spit out very concrete numbers right? So whatever we're doing on the string theory has to give you those numbers. So first of all they're finite. Okay that's also a non-trivial thing to get out of the string theory. So we're going to get out some finite numbers and in particular we're going to have to be very careful about this little bar over here. So this bar is basically boundaries of moduliz space. Okay. So when we do integrals as you know sometimes we integrate by parts it's very important that you get boundary terms. So it's not really strictly speaking a boundary there's going to be actually sort of several layers of higher and higher co-dimension parts to this moduli space which are sort of all the different ways that these remon surface can sort of degenerate. Okay so you already saw one example here where we have this remon surface. So these are sort of most of what the modulized space of the M11 looks like. But there's a co-dimension one part where basically one of these cycles shrink. They pinch. Okay. So in the language of tao and ta bar you might be familiar with of the taurus this is at the point toao goes to i infinity. Okay. So we need to keep track of some special things that can happen at the boundary. And what's really amazing is that there's sort of this mathematical technology that allows you to actually have an algorithm you can put on a computer and that's going to spit out this integrant. So what I want to do is to take the string duel to say the quartic matrix integral and spit out these integrants for you and it's very very concrete. So what this first line is is just importing all the packages where all the hard work was done. Okay, these strings. Here we go. Okay. sort of a little factory that creates sort of these topological strings for you that's going to allow us to kind of compute everything. And for some low genist Is that how about now? Okay, there we go. So what we're able to do already is so for the first few ones that Reesh mentioned in this first lectures we were able directly to compute this by hand. Okay but I just want to do some checks. So what we're going to do is we're going to look at these correlators. So the case of the genus 03 point function the moduli space is just a point. Okay so when you want to integrate something on a point that's just saying that the integrant is the final answer. Okay. So here what we're going to do is we're going to in the quartic model. So this one here is telling you that we've turned on this T4 that Reesh has spoken about. So there's a quartic potential in the matrix model and we've now just computed literally directly these correlators as a full function of the tof coupling T4. Okay. So it's resummed infinitely many fineman diagrams and it just spits out the answer as a function of the tooth couplings. Okay. So this is here for example you can see this would be trace m^ squ trace m^ squ trace m so there's an odd number of traces in this in this correlator and so in an even potential that has to vanish just by symmetry from m goes to minus m right so indeed the string theory is smart enough to know that and it gets a zero here and you can compute some other correlators okay so we can double check that this is indeed what we see so for example this is where all the kis is equal to one okay so let's see this gamma to the 4. So this is 2 * 1 + 1 + 1 3. So there's a 6 - 2 that's this gamma to the 4. Okay. And this 2 factorial 2 factorial 2 factorial that's this 8. Okay. So the string theory is indeed reproducing what you saw in the very first lecture as computed from the matrix model. And then we again recomputed directly from the string theory. Okay. But let's now move on to this genus one one point. Okay. So we can compute a few of these correlators. Okay. So first of all we're computing trace m to the to m to the k in a even potential. Okay so that has to vanish whenever k is odd. Right? So indeed the string theory again knows how to do that. So that's this v1 v3 v5 that are all zero. And here you're recomputing these correlators again as a full function of all the tooth couplings. You've resummed infinitely many fineman diagrams. Okay. But the way that we do this is to really integrate a particular form on this modulized space. Okay. So for example in this genus one one point function there's a sum of three terms. Okay so the coefficients aren't important for you right now right but the point is that there's going to be three terms. So what these s ones these kappa ones and this stuff is they're just particular differential forms. You don't need to know what they are but there's some particular closed differential forms. But more importantly is that people know how to integrate them. Okay so these are things that are very well studied in algebraic geometry is that all the integrals of these things are known and they're finite numbers. Okay. So that means that once we have this integrant on mgn we also know how to integrate it. Okay. So let's look at how we so this is the code that for example generates this uh this integrant. Okay. So what we're looking at right here is for uh trace m^2 and we want to find what is this integrant v2. Okay. And so you can see so on most of the moduli space this is what this integrant looks like. Again this kappa one and the s1 is just some particular twodimensional form. So remember the moduliz space of a of a taurus is two dimensional. So you're integrating a two form on it. Okay I'm just telling you that there's these two combinations of two forms s1 and kappa 1 that you're integrating directly on m11. But actually there's a special part at the boundary okay precisely when these taurus when sort of one of the cycles of the taurus pinch and you have to be very careful. Okay, but what's amazing about this formalism is it also knows how to keep track of those boundary terms and that's what this little graph is here. Okay, this 01 is saying you have a little one here and you can think of these two points as two and three. So that's what that two and three is. Okay, so that's this degeneration of this Taurus and this is telling you this is precisely the integrant that's localized on the boundary of moduli space for the dual of the string. Okay, of the cordic model. Now, usually it's almost impossible to do anything beyond genus one. Okay, so this is why it's very nice to have a code that allows us to do things. So, let me show you an answer. So, what we're going to look at is the integrant dual to trace M squ, but we're now just for the fun of it going to do it at genus 2. Okay, so let's do this. So now, so you can see it's having to do a lot of stuff. This is going to run for a little because it needs to take this genus two two surface and it needs to think of all possible ways that this thing can degenerate. Okay, so for example, you could pinch one of these cycles. So it might look something like this. Okay, or you could pinch one over here. So it might look something like this. Okay, and it's going to do all the various combinatorics. Okay. And on each one of the parts of the moduli space, it's going to spit out some very specific integrant again as a full function of the tooth couplings. So this is again to all order than the tooth couplings. So for people who maybe think about ads, usually the best we can do for computing something from the world sheet even at genus zero is maybe two orders in one over the tooth coupling. Okay, this is genus two all orders in the tooth coupling. Okay, so this is what these look like. So on the big part of the moduli space that's what the integrant looks like. Okay, this guy over here is precisely this guy. So that's this genus one one point where now we have this two and three. Okay, so it's it's a pretty non-trivial integrant. Okay, but you keep doing this. Okay, so all of these boundary terms that it's actually computing for you. Okay, it's very very non-trivial. Yeah, so there's a lot to do and now we still have to do the integral. But I told you all of these differential forms mathematicians know how to integrate. they've already done the work for us. So it's just specific integrals with specific coefficients. So we can just do this integral and that's so much nicer. Okay. And in particular, so this is trace m^ squ at genus 2 and this is something you can compute from these nonperturbative schwinger Dyson equations in the matrix model. And you can check that that's indeed the right answer, which we won't do right now just because we're a little out of time. But we can do different things. So as one of the things Reesh was saying is that if as he explained to you yesterday if you look at correlators in the Gaussian matrix model the correlators are just counting with contractions right and we keep track of them by drawing these fineman diagrams. Okay, so what should come out at the end of the day are some integers. Now, as he was just saying, integrating some random forms on mgn and at the end of the day, getting integers is a very non-trivial thing for a mathematician, right? So, let's just do this. So, we're going to do the genus one point function. And this actually reproduces a result known to mathematicians in 1986. So, they actually tried to look at the comatorics of these diagrams and they figured out these comtors at genus one. Very non-trivial problem. Okay. And indeed you see that all of these coefficients in front are integers. Okay, so it's a to I mean you saw the type of integrams we're dealing with, right? There's a lot of coefficients in front. The various integrals are also not integers but at the end of the day everything resums up into these integers and these integers are precisely counting these number of witch contractions. Okay. So um you can do your favorite potential. So uh here we thought just to show off we would try something with quartic sectic octic. Okay. And we could uh so we can build a little uh chromological field theory. So we turn on this time t uh t4 t6 t8. So let's kind of run this. And now we're going to again compute so at genus one a twooint function and look at the integrant. Okay. And so again here you go and it's a function of you see this T4 uh the T8s appear over here and it again tells you on every parts of the modul space what the integrant is. Okay so you can see this might seem all very very abstract but it's something that you can boil down to to putting on some Mathematica code. Well it's not really Mathematica. It's a lot more involved but uh um so it's at the end of the day it's spitting out for any correlator in any matrix model that you want any genus any endpoint function it's giving you a particular integrant that you can now integrate on MGN and that integrant is totally non-perturbative in the string couplings okay it's reummed all of the perturbation theory of the matrix model which is what the whole idea of string theory was for okay in the beginning days of toft they were trying to find a string theory to describe strongly coupled gauge theories because the string theory would resum the coupling expansion for you and this is precisely what you see. So I'll maybe stop here and then if you have any questions about what we were doing or if there's some particular correlator you really want to know the answer to I can also give you that. So thank you So very impressive. Very impressive. Uh just one comment. >> This is really Alessandra's work. Huh? >> Yeah. Yeah. Yeah. No, no. I I know that they have been also working on other codes for topology. >> Yeah. There's a whole package of chromological field theory due to a whole workshop of mathematicians that's running behind this. Yeah. My question is about the relationship to other approaches because you know you remember you know you're having this duality between the one matrix model and and string theory but the old work of diagramraph and buffer was proposing that one matrix models were dual to some special calabio but one question I I I I never fully understood is that you know in a sense there was no independent definition of what this topological stream theory was on this special calabas right this special calabas for example didn't have a name model realization right >> so in a sense you seem to be giving here a explicit construction of a B model >> of a B model so I would also like to know the answer to that I should have mentioned that there is this proposal of diagramrapha and uh I think later also elaborated with Albert and you and Mina and so on so but I think as you say it was a very nice geometric picture based on the sort of the brain construction. But I also the kind of detailed operator dictionary uh was something that was not very clear at least to me and how you would be able to uh to uh to compute and I I think it may not be unrelated. I mean the the spectral curve is kind of behind them but this is more in the Lando Ginsburg description and there's maybe some way to I think there are these Lando Ginsburg Colabio equivalences in some cases so uh so it may be there may be some way to map it to that but uh so I don't know the precise answer I should say but uh yes so that's why there I felt I at the world sheet level I didn't have a real understanding of what the theory was. Uh >> yeah I mean I mean in in that picture trace these these correlators will be open string correlators already. >> Yeah there would be some brain so then it would be just the brain. So of course what we did afterwards was to consider uh backgrounds which have an a model dual and then you know we did this remodel in the B model using topological recursion but uh you know so it's a theorem by now but um you know so there should be I mean of course you know then you know the matrix model in a sense is what is not obvious there right you have the you don't have a specific matrix model description what you have is the spectral curve description uh but you know in I think that part might I mean of course Alexandra knows about this uh remodel and the remodel very well. So the rem the >> Yeah. So, so, so maybe you know this is also kind of a another but I I I would say that you know this this really gives a very nice commical filter reconstruction of this B model in the casing where the calaba really is not an standard calaba right >> because for this diagram of a park the calabao the calabaya it's it's really very you know I mean I wouldn't know how to compute the topological string on that calabao from other means that were not using the matrix model directly You see? >> So this in a sense gives a seems to give a very kind of explicit description of those backgrounds in a sense. >> Yeah. And maybe there if there's a way to map these to those calabios that might help to give an independent definition of those calabios at least how to yeah maybe one thing I'll also add just to that is that the so in this case it's really just the spectral curve that's the target. it's not somehow embedded in some calabial three-fold. And because you're not in that three-fold critical dimension, that's why also these gravitational descendants play an important role. >> So here you really just have sort of a two real dimensional target >> and you have to sort of face the fact that you have to deal with these gravitational and sort of that's what this is doing. >> Yeah. So it's not part of non-compact three-fold in some sense at least not obviously but maybe it's some reduction of that or something like that you can uh topical gravity I mean this is this is still you know in this you know when you do a topic on this calabio >> well of course you know the B model philosophy is not to do integrations over the model space in a sense it's slightly different it's used to the formation of complex structures but uh but yeah I I see your point but there there should be something similar to gravitational descendants. But yeah, I I mean I really like a lot this picture and I think it's really filling an important gap in our understanding. So that's my kind of >> yeah and the dual in some sense there's also a part which is just the holography I mean matrix models are zero dimensional the and there's you would expect that the dual is sort of one complex dimensional or the holographic dual is and which is what the spectral curve is. So in some ways there's a nice picture I think uh maybe you want to say I mean about how you can think of uh the the matrix model as sort of living at infinity in the spectral uh curve and uh and but the string theory side giving contributions from the critical point. So zed equals to infinity is sort of where the matrix model lives but here zed equal to plus minus one is where you get then the the string contribution. So there's some kind of a interesting holographic uh picture there. So it's more in line with your stand with your standard expectation of holography. I think you mentioned that uh you have topological recursion on both sides of the duality. So that that sort of makes you know uh the equivalence more explicit. Uh now on the string theory side is the topological recursion is something like the Mirzakhani formula on moduliz spaces or uh is that something formula? No. >> Is that what you said? >> Uh but not not the not the exact same formula. It's not the exact same formula >> but people had studied these topological strings in the 90s and Whitten had sort of derived these early recursion relations for the correlators genus and genus one >> and Berinda and Verinda in the early 90s showed that for example the correlation functions just of stuff in sort of the gravity sector satisfied a recursion relation in all genus >> we're able to comput things that way >> I see >> there's a there's a way to understand where is um there's essentially a way to kind of localize this integral sorry the integrand to the boundary of moduli space >> and it turns out that the boundary of moduli space of mg looks like other copies of mg >> okay >> turns out because of that you get a recursive structure so that's roughly the origin >> I guess that's those are the things that >> and that's roughly what's also underlying this technology over here >> I see I see okay I I'll ask other questions later thanks Are there other questions? So um I would have a question uh regarding the functions you have shown us as function of T4. So >> the function of >> the functions you have shown as in the in the file as functions of T4 because so the matrix model is basically a latis generalization of what Marcos has told us in his lectures about the zero dimensional 5 to the 4th uh path integral which uh should give an analytic function when the coupling equals to zero. But here you're showing uh rational functions. So I would assume you're basically summing over a convergent series. So like what am I missing here? This is a remark that Marcos made yesterday that at each order in the genus expansion for the models we often consider the tooth converg the tooth expansion is actually a convergent one >> and that's an aquatic model there's this 1 by 12 after which it will sort of become like this I mean it becomes inverted and so >> you kind of >> yeah Right. >> How is that grow factori? >> Okay. >> And you see the point that Marcos mentioned yesterday. You can always you can see in all of these things that all genus it diverges at the same point at 1 by 12. I mean genus 2 genus one all of these answers have the same divergence. Yeah. Okay. Uh so maybe this will be the the last question since we are a bit behind. Yeah. >> Very short thing. So uh uh so you mentioned that uh like these forms mathematicians know how to integrate. So is it simply that you get a very big answer for the form but then you can sort of analytically integrate it and it becomes something very small. It's just that or is it >> and basis of forms where you know >> okay >> then the coefficients might look a little complicated >> and showing you that at the end of the day once you everything up you get a nice answer >> and you also said that it's like localization to the boundary of the modiz space. Basically it's just that you integrate and evaluate it at the boundaries and >> no that was just for this particular proof strategy. >> Okay. >> But uh >> but still analytically doable like Yeah. Okay. >> Okay. Great. So I if you have more question I invite you to ask in private to Rajes and Edward. Uh so this was the last lecture of the series. So please thanks uh the speakers for that.