 Let me discuss first the G-hat animal. It has to do with the Gromov-Witton theory of X. So here, first let me tell you about G without the hat. G of t is a power series, some c, well, my cd, cn, t to the n. En is essentially a Gromov-Witton number. Let me just write first and comment later, 0, 1, d. OK, let me just write this then. Sorry, n. Side dn minus 2 of upper star point. What is this? So here, x01n is the modular space of maps, morphisms, f, from a genus 0 curve with one marked point. OK, so that's a genus 0 curve. For example, it could be p1. So that's the first 0 here. It has one marked point here. x is one marked point. That's the one here. I'm mapping to x such that it has anti-canonical degree n. So the degree of f upper star minus kx equal n. And you see, it's always strictly positive because x is final. So this is always strictly positive as x. This thing is out. And let me go into the details of this. It's not clear that I will discuss this material, perhaps towards the end of these lectures. I'll tell you a bit more about these things. This is the conceivage space. There are some technicalities in compactifying it, and so on and so forth. And then here, I'm really integrating on the virtual fundamental class of it, which I'm certainly not going to talk about. But let me tell you about this psi thing. Psi is the first churn class of a line bundle on that space. Yeah, sorry. First, perhaps I should say the evaluation map. This thing has an evaluation morphism to x, which takes the map f to the evaluation at the marked point. So that allows me to pull back to this space the homology class of a point, the general point on x. If you like, I'm looking at all the curves, all the p1s of degree n in x that pass through a general point. Psi is the c1 of l, where is a line bundle on x0 is the line bundle on x0, 1n, which with fiber at a particular map, the cotangent space of the curve gamma at the marked point, x. Little x, that's a little x, the marked point. And these things are doctored so that the degree of that homology class is 0 and then this is a number. cn is a number. That's the g. And the g hat is a similar power series where I multiply all those cn's by n for c2. In particular, if you haven't really much seen this stuff, you should not worry, OK? All I need you to take home at this point is what kind of information this is. This is some functions. It's a generating series whose coefficients have to do with rational curves of degree n on x. It's a ground-footen-theoretic object. That's the left side, OK, of Merosimacht. What about the right side? So what's pi w? And pi w is a classical, much more classical object. It's a standard period integral of a differential form on a cycle. And here gamma is some cycle hn of y with integer coefficients. And in various situations, you would want to choose it. But typically, there is only one such cycle that makes sense to consider. And omega is a normalized volume form, n form, holomorphic n form. Omega is a holomorphic n form on y, typically normalized such that a little what? Bigger, bigger, OK. Yeah, OK. Yeah, yeah, makes sense. A holomorphic n form such that the integral over gamma of omega equals 1, OK. OK, so anyway, in these lectures, yes, we'll be concerned with the low-level interpretation of Merosimacht. Very sized cycle. And typically, there is only one reasonable cycle that you can consider. Why don't I understand? I understand. Why don't I make this concrete at least, OK. So why don't I give you an example, OK. So then it's clear, or at least in that example, what we're talking about. Merosimacht is not exactly science as such. It's more like a building site, OK. And all these questions are completely legitimate. So example, p2. So x equals p2. So here I want to take the mirror to be y is just a two-dimensional torus, c star squared. And w, I want it to be the function x plus y plus 1 over x times y, yeah. And the standard cycle gamma on y, I'm going to take the obvious thing, s1 cross s1. Inside y, that's the set. If you like, norm of x equals norm of y equals 1. And omega is the invariant differential form on the torus. And if I want to normalize it so that it has integral 1, I better divide by 1 over 2 pi i squared, OK. And so what's the period pi w of t as just the integral over gamma of 1 over 1 minus tw omega? And so I can compute this with the Cauchy's theorem. So I expand this thing in power series. And so that's the sum t to the n times the integral over gamma of w to the n omega. And by Cauchy's theorem, can you still read despite this monster object here? Am I supposed to stop at some level here? I am supposed to stop some level here. So then that is, by the Cauchy's theorem, the only thing that matters here is the coefficient of 0 in w to the n. And so then I'm talking about some t to the n coefficient of 0 in w to the n. And so when you think about it, what's that coefficient? So w to the n is x plus y plus 1 over x y to the n. And you're going to have a constant term only when n is divisible by 3. And then your constant term is going to have to be some sort of binomial coefficient. You want so many of these exactly as many of those and exactly as many of those. And so that's 3n over nn t to the n, t to the 3n. OK, so that's your period. On the other hand, then let me continue over here for x equal p2. What is the function g? Well, that's actually deep. This is the work of Given-Tall. Given-Tall tells us that g or p2, g of t is some t to the 3d divided by d factorial cube. That's, indeed, the g hat equal pi. So I'm debating with myself where I want to take these questions just slightly further. Maybe I can. These period functions, I w of t satisfy algebraic differential equations. These are called the Picard-Fuchs equations. And there is some general theory that tells you that. And similarly, these g hat functions also satisfy some differential equations. And that comes from something that happens in quantum cosmology. And another way to state the low level, a low level mirror symmetry statement, though not as low level as this identity of these two functions, is to say that some regularized quantum differential operator here that annihilates the g hat function equals the Picard-Fuchs operator on this side. And it is sometimes helpful to state mirror symmetry in this way. And all I want to do here is to compute the differential operator in question, or P2. I like to write my operators as, in this form, some t to the k, I'm saying polynomial in d. So my sum will go from k equal not to r. And here d is t d by dt. And so this operator kills a power series. So suppose that phi of t equals sum c and t to the n. So such operator kills phi of t. Sorry? I am right. Well, in principle, this works for both sides. I really only work with a g hat. Yeah, a g hat. But in fact, I'm graded on the pi here. Well, whatever. The thing that has that formula there. And so l times phi equals 0 is equivalent to a recursion relation between the c n's that reads, I mean, please let me just write down the formula. You will work it out in a moment. That's c n minus k pk n minus k equals 0. And you probably want this for all n larger than or equal r. OK, thank you. The interpretation being that all the negative c's are supposed to be 0. And so for p2, pi equals 0 for l, the operator d squared minus, OK, let me copy it so that I don't make a mistake. 27 t cubed d plus 1, d plus 2. And you can work that out easily from the form of the coefficient, this form here. So that's 3n factorial divided by n factorial cubed. And you see that nth one of these is expressed in terms of the n minus first of these multiplied times 3n minus 1, 3n minus 2 divided by n cubed. You work it out, and that gives you that particular operator. So another way is another low-leverant interpretation. Mirror symmetry is an identity of two differential operators. And on a slightly higher level than you might want to say that actually it's an isomorphism of d modules and so on, you can layer it progressively higher and higher levels. I will use both the functions and sometimes the differential operators. Anyway, so that's an example. OK, we spoke about p2. We will have many other examples. Today, I'm not going to tell you what we will do. I'm going to tell you next time. But I want to tell you somehow what the point of view here is going to be, some of our point of view. So as I said, mirror symmetry is not a finished theory. It's more of a building site. And so here are some general, completely open questions. So question one, how general is it? For instance, do all phano-orbicles smooth? OK, so we don't know. I know that for all the known phano-manifolds. The answer is yes. Well, OK, so that would be a slightly exaggerated statement. But not that far from the truth. I know very many phano-manifolds, and I know their mirrors. But I also, in my experience, I don't know any counter example for smooth-orbicles. So we've looked at the edges of what we know, and it seems that we can always make the mirror. Yes. Yes. Yes. Yes, yes. Yes, yes, yes, yes. I have a paper on this with other people. We will talk about this precisely the next time, OK? I will try to, what's the word, make you happy. OK, so that's a question. Question two, how precise is it? So what do I mean by precise? I mean, can we define two sets? One set having to do with phano-orb, before the let me call it F, OK, and another mysterious set, let me call it P, which might have something to do directly with the Landau-Gaswell model, or maybe something else. And then there is a one-to-one invertible function for respondents from F to P, such that then if I take something here and take F of P, then all possible low-level, high-level mirror symmetry statements hold for that particular pair. And can we state it in this form, where the sets F and P are going to be exactly defined in terms of things, and we can make sure that one is an element of F and one is an element of F. Question three, OK, is it conceivable that we can make, can we make a directory, maybe a computer directory, eventually a complete classification, phano-LG-PASS? So in other words, and here I mean something very effective and very practical. You know a book, like the yellow pages of phano-LG-PASS. And then you have to imagine what the data set structure is going to be here. There are some phone numbers in these yellow pages, and I can look for the phone number and use the phone number to ring up the phano and know that somebody will answer that phone. And then I can rig up the LG thing, its mirror, and I know for a fact that somebody will answer that phone and that this pair, this thing, is mirrored to that thing. And if you give me a phano, then I'm going to ask the directory to put me in touch with its LG model. And it's going to be explicit and constructed, and it's going to be as a data set inside this directory. So that's the question. Completely practical, OK? So what we are about in my group and people I work with, Al, who is here, we're about question three. We want to make this directory, OK? And so I just want to, and there's even more of a building site and working progress for us. So we cannot tell you the answer to this problem in this course, because we don't know about it. But whatever we do in the course, we do because this is what we want to do, OK? So when we do something, then you know why we do it in that way and not in some other way and you don't come and complain to me. So you see, because I can well imagine that there is a beautiful answer to question one and two, which is completely useless from the point of view of doing question three. Somebody may come to me and tell me that, oh, yeah, I discovered this theorem. And the mirror of the Landau-Gissel mirror of x is some modular space of stable objects in the derived category of x defined on this stability condition or whatever. And that's great. I mean, it's a great and extremely satisfying theoretical statement. But I want the phone number. I want to call up that Landau-Gissel model. And I want it to turn up as a data set that I can make the damn thing, OK? So to know that it's something like that, it may not help me at all. So it's not that we don't like theory. We love theory. But I'm just telling you, because that's what we want to do, we are interested in the kind of theory that helps us answer question three. How long are these lectures supposed to last? One hour. OK, so very good. So anyway, so in terms of context, I've told you more or less what the theory is for us, the low-level interpretation and what our point of view is. Next, or this is going to be mostly the working next lecture, I wanted to say what we are going to try to do in this course. And in order to tell you that, I will sort of continue on. I cannot right away tell you what we plan to do. For that, I need to start with some preliminary discussion. So obviously anyway, in terms of question three, we are interested in explicit constructions of merosimetry. So starting from a final given with equations, how can I make the LG model? And I want to tell you about one particular constructions. So one of these is the Horivafa construction. This is described in this paper called Merosimetry, republished from 2000. And it's in the archive, so you can pick it up. And if you like, you have to look at section seven and particular equations 7.83 and 7.84. But it's been discussed in many other places. So if you don't mind, I want to spend some time telling you about this thing. This is for Toric complete intersections. So by the way, I intend to spend a couple of hours later, maybe next week, telling you about Toric geometry in a way that plays well with the kind of stuff that we do. How many people here have seen the basics of Toric geometry before? You see, that's good. It is almost everybody. But again, not everybody raised their hands. And I don't want to talk to the people who ought to be teaching me. I mean, this thing is to make sense. I have to talk to the people that I'm sure that I have something to teach too. And certainly, those people who did not raise their hand are such people, for example, concerned. So I will spend some time talking about Toric geometry. But for today, well, it's not that I'm going to go very far with this. From the beginning, what's the Toric variety? What's the Toric variety? If you open up a book on Toric geometry, they start with some crazy definition that says the Toric variety is already X with an action of a torus on it, such that a bunch of things happen. It's a completely ridiculous thing. Nobody wants to know what the Toric variety is in that way. And it's extremely inefficient to using the books to what's the data set that's appropriate for telling yourself what's the Toric variety. That's completely inefficient. So for me, a Toric variety is just a quotient. So it's a quotient, cn. Well, in fact, it would be always an m. Modulo in a r-dimensional torus. So here, tr is the spec of cl. And where l is the rank r lattice. So typically, and if you want, you can always imagine that l is isomorphic to zr. But occasionally, it's helpful also to allow l to own some torsion. But that's an additional technique that's not terribly important. Where tr acts on cm as the linear representation. So a vector space is cm, the linear representation of a torus. And I want to take the quotient. And really, you know an example of this thing, the projective space. It's cn plus 1 modulo, the action of c star. And you already know in the case of projective space that you have to be careful. Before you take the quotient, you have to throw away the origin. But then after you throw away 0, you can't take the quotient. So what happens here is that there's something subtle going on there. For tori of higher rank, there are actually different choices of taking the quotient. And the stuff you throw away, it's debatable. And I will tell you a little bit about this. You need to choose a stability condition that for today, we're going to ignore. But so then, this is a geometric quotient of this by this taken with respect to a stability condition. And you probably want to put it in square brackets because you want to think of this as an orbit fold, not as a just variety. And for me, torque varieties are always denoted by F. But in principle, this is just not much harder than a projective space. And I'm going to demonstrate this to you by taking some of these things and carving them with charts like you cover projective space and just show you that you can learn to live on such a thing. And it's not going to be that much more painful than living on projective space. And we'll continue the discussion next time. But to wrap this up, then how do you give yourself a torque variety? Well, you have to give yourself a data set variety. So you want a group homomorphism from ZM to L, OK? So which dualizes indeed to a group homomorphism from our dimensional tors dual to L to C star to the M, which acts diagonally on Cm, OK? And then you measure of taking the quotient. So that's how you give yourself a torque variety. And let's not worry about how you choose your stability condition, OK? And then you want to give yourself line bundles on it, C line bundles, L, C on F, OK? And a line bundle is essentially a character of your torus. So these correspond, these are basically just elements. These are one of L, C are basically just elements of L. So those characters arise to define an action of the torus on Cm times another copy of C. And when you take the quotient, you act on Cm with D, and you act on C via that character. And you take the quotients, that's a line bundle on F. What does it mean to say that X is a total complete intersection? Then X is the intersection of the variety of F1, intersection the variety of Fc, where these Fis are general sections of these line bundles on F, OK? Then this X is a toric complete intersection. It's X. Sorry. X is a toric complete intersection in F. Yeah, so that X is a complete intersection in F, OK? So here I'm choosing C line bundles on F and general sections of them, and X is just the place where all those things are 0. So tomorrow, then I'm going to tell you when I have one of these toric complete intersections, which happens to be fun, oh? Now that some silly assumptions are satisfied, then they give you something that's supposed to be the mirror of it, OK? We'll talk about this tomorrow.