 parametrized by either a single number z, and I had an example which was just a family of cubic polynomials, or parametrized by two numbers, z1 and z2 and we had a family of quartic polynomials like that. In any case, once we had this family of polynomials, sorry, yeah, it's okay, it's just a screen thing. We had this family of polynomials and from that we produced, well, ultimately, we produced some concrete gadget called spectral network drawn on the parameter space c. It's a network of co-dimension one curves, co-dimension one general walls on c, depending on the choice of a phase which we call theta. In the end, what I gave you was just a concrete construction that just starts with a polynomial and produces this thing, but the way we were thinking of it was that starting with the polynomials, you produce something way more elaborate, which is an n equals two comma two supersymmetric quantum field theory. In that quantum field theory, there's this supernatural question to ask which is, what are all the BPS states? Or, as we said last time, a BPS state comes with this kind of funny invariant associated with it, a number, which is a complex number. The mass is equal to the absolute value of this number. And so the slightly more refined question was, what are all the BPS states that have phase, where the phase of the central charge is equal to theta? And so this picture is supposed to be the answer to that question. It says, for any point z, you ask, at that point, is there or is there not a BPS state whose phase is exactly theta? And if there is, then you color that point black and otherwise you color it white. So this is the picture which I also drew last time. In the case of the Cordic, where I fixed the parameter z one equals to minus one, and I'm just letting z two vary in the plane. Well, to be exact, this is the picture at theta equals zero, if I remember, right? I know, this is the picture at theta equals pi over two, I think. And I wanted to show you what the picture looks like as you change the phase, partly for fun and partly it'll be important later. So first of all, one thing you notice is, well, okay, let me just show you how it moves. So as we change the phase, the thing moves continuously. And in fact, all through all the values of theta, it just moves continuously. There's never a kind of discontinuous jump in this picture. You can convince yourself it kind of had to be like that. Okay. All right, maybe that's actually all I want to say about this right now and we'll come back to fancy ones later. So, okay, so that's a sort of picture that captures the spectrum of VPS states in this quantum field theory. Now, you might reasonably ask, is this picture good for anything else? So I want to tell you one other sort of mathematical thing that these pictures are related to. And for that, I need to introduce a little bit more structure. So let's talk about chiral rings. So yeah, it'll just be some concrete mathematical construction but let me to put it in the sort of broader context. So if you have a space C, which is the parameter space, really I should say the chiral parameter space. Anyway, some kind of parameter space of the quantum field theory. Parameter space of N equals two comma two theories. So these C's are examples of that. It carries a holomorphic vector bundle. It's a complex manifold. So we could talk about a holomorphic vector bundle that carries a holomorphic vector bundle and not only a holomorphic vector bundle but it's a holomorphic vector bundle of commutative algebras. So each fiber of this bundle, if you like, is a commutative algebra. Call it E. And those commutative algebras are called the chiral rings. So it has to do with the multiplication of operators in the topologically twisted version of this theory. But we won't need to know that. And the other thing it has is a map from the tangent bundle into E. So every, and the way you should think about that is that the quantum field theory sort of knows its own deformations. If you wanna deform it, the way you deform it is using one of the operators of the theory. And so every deformation, in other words, every tangent vector to this parameter space corresponds to some operator of the theory, something in E. Okay. Now, although it wasn't said this way in the early literature, there's another way of rephrasing this data. Which is, let's define phi, a map from the tangent bundle to C to the endomorphisms of E, a holomorphic map, by, okay, so phi of, so for V a tangent vector, for V a tangent vector, I have to get an endomorphism of E. So I have to get something that can operate on a section of E, okay, let's call it little E. So in other words, what I need is something that uses a tangent vector, an element of E gives me another element of E. Well, I've got a candidate because what I do is I take Q of V, now that's an element of E. E is an algebra, so I just take Q of E dot E using the algebra product. Okay. So the holomorphic vector bundle E comes with this extra gadget phi, and that's a gadget that you've seen, I guess many times by now, namely it's a Higgs bundle. So E comma phi is a Higgs bundle over C. So I'm saying these parameter spaces, the spaces where I'm drawing these pictures are also kind of canonically the basis of some Higgs bundles. And so it's natural to ask is there any connection between the Higgs bundle and that picture? What does that picture tell you about the Higgs bundle? That's what we're gonna try to sort out. Okay, so, but first, this was some generality, but I gotta tell you now how to make this Higgs bundle in the particular case of these families of polynomials. What's the Higgs bundle gonna be? Okay. So in, oh yeah, this two comma two supersymmetric field theory was called the Landau-Ginsberg model, right? LG model. LG model. So in the LG model, supersymmetric Landau-Ginsberg model, these algebras are pretty simple. It's just you take all polynomial functions in one variable, the variable X, and then you mod out by the ideal generated by the derivative of W. In higher dimensions, I would mod out by all the derivatives of W. Here, W, I think of, sorry, yeah, this is for, so W depends on a parameter Z. We're not taking the derivative in that parameter. Think of that parameter as being held fixed and we're taking all functions mod out by the X derivative of WZ. And then the other thing I gotta tell you is what is this map Q? So, and Q acting on some tangent vector, which I'll just write generically as D by DZ. So that, I gotta give you an element of E and that'll just be the Z derivative of W. Okay, let's see how it works in an example. Absolutely, so one thing that's, yeah, so if you have a family of field theories, so yeah, one thing it means is that indeed that all the correlation functions, even the algebra of operators varies in some kind of locally trivial way over the parameter space. In this case, the, yeah. No, not everything varies holomorphically, but if we were to restrict to the topological part of this theory, so it's an N equals two, two, supersymmetric theory, you can make a topological twist of it or said otherwise you can look only at the operators that are annihilated by some supercharges. If you look at only correlation functions for the special class of operators, which are exactly the ones in this chiral ring here, those correlation functions do vary holomorphically. Yeah, well it's hard because I haven't said what I mean by A theory. So how could I reasonably do that? Certainly, right, if you think of quantum field theory is defined by a Lagrangian, then indeed you could literally just explicitly deform the Lagrangian, make the Lagrangians be functions of an additional parameter. I mean that's exactly what happens here. In the Landau-Ginsberg model, in the Landau-Ginsberg model you literally write down an action and sort of the main non-trivial ingredient in that action is this function w of x. And so here we're just explicitly putting a parameter in that w of x. Don't make me try to do it in real time. Anyway, sorry if this is not the kind of answer you want, but maybe I can try more offline. Yeah, that's right, that's right. So the question is why don't I deform it in some arbitrary way? Yeah, so the statement is that when you make a deformation by adding a chiral operator, well maybe the simplest thing to say is the correlation functions in the chiral sector, the correlation functions of the operators that are annihilated by the Qs, it is a fact that those correlation functions depend holomorphically on the parameters entering in the super potential. Yeah, I said otherwise, maybe a better way of saying it is the following. The Lagrangian is constructed from this super potential, but it's constructed in kind of a specific way. Varying the super potential is not the same as making a totally random variation of the Lagrangian. So the statement is that the correlation functions I'm interested in depend holomorphically on the coefficients in the super potential. For a random deformation it wouldn't be so. That's what I meant here by saying chiral deformation. But it turned out in fact, see the cool thing is that if you make a deformation and you want it to preserve the supersymmetry, if you want it to be a deformation which keeps the theory n equals two comma two supersymmetric, there aren't that many ways to do it. There's basically two ways, chiral and twisted chiral. One of them is this kind of deformation where everything is automatically holomorphic. Okay, sorry. Let's talk about it later. So yeah, I wanted to write concretely what this Higgs bundle is. So let's do that. Where's the eraser? So in this cubic example. So to write this in just an absolutely, well let's see, all right, let's do this. So EZ is C of X mod out by W prime of X which in this case is just X squared minus Z mod out by that ideal. So it has a trivialization, or EGZ has a basis so it gives a global trivialization given by say one and X. Let's write X squared, X squared is already equal to Z. Then relative to that basis, so I've got a rank two trivial holomorphic vector bundle over my parameter space which is C. And then the only other thing I have to give you is the Higgs field. So let's write the Higgs field. So the Higgs field acting on the tangent vector D by DZ is well that's supposed to be the Z derivative of W acting by multiplication. The Z derivative of W is minus X. It's minus X acting by multiplication. If you work out how minus X X by multiplication in this basis you see it's a little two by two matrix. It's minus zero one Z zero. So concretely there's your Higgs bundle. The Higgs field is this times DZ. Yeah, thanks. No, no, no, I meant five D by DZ. I'm giving you already the Higgs field using that formula but this is the formula for the Higgs field. Okay, and similarly in the quadratic example, okay by similar computations which you can see in the notes but I'll just write the result. So here E is isomorphic to C three with a basis one X and X squared and the Higgs field. So now this is a Higgs field on a higher dimensional space, right, this Higgs field on C two. So phi of D by DZ one comes out to be this matrix. Zero zero one Z two Z one zero zero Z two Z one. And phi of D by DZ two is minus zero one zero zero zero one Z two Z one zero. So this is a Higgs bundle over C which is C two. Now actually in defining a Higgs bundle over a higher dimensional space I guess there's one condition that probably hasn't been mentioned explicitly because usually we tend to talk about Higgs bundles over one dimensional spaces. In higher dimensions there's an extra condition you put which is the bracket of phi with phi is zero. So concretely what that means is that this matrix should commute with this matrix but that's true because the chiral ring is commutative. These are all multiplication by different elements of the chiral ring. So we automatically get an honest Higgs bundle over this higher dimensional thing. Okay. Okay so so far I told you about a Higgs bundle, some simple canonical Higgs bundle that exists in this world. I didn't tell you yet what it has to do with the picture I drew. To tell you that I have to introduce yet a little more structure. Yeah. That's a reasonable, okay I just haven't thought about that. Presumably the reason why you asked this question. Yeah so there's an interesting question I wanna answer it to make a point. So Ray asked why is this Higgs bundle stable? And I think what he has in mind is that there's a solution of Hitchin equations on this Higgs bundle just if it's stable. I didn't say anything about Hitchin equations yet. It's true that this bundle is gonna carry a solution of Hitchin equations but this kind of quantum field theory way of thinking of it is gonna produce that solution directly. You're not gonna get it by solving some PDE. I mean it's true that it's a solution of the PDE but you don't rely on some existence theorem of solutions of the PDE. Yeah in a way I sort of don't care whether it's stable but it must be stable because it has this solution. Yeah, okay so I wanna tell you a little more structure that it has. So this Higgs bundle maybe is cool but maybe you would say not super deep. Then there's some more structure in this story which is a little more interesting I think. So it's also supposed to carry a symmetric C bilinear pairing. Yeah. And a family of flat connections which I'll call NABLA H bar. So they're gonna be parameterized by H bar in C star complex number but they're of a particular form. There's one connection called NABLA infinity and then you shift it by H bar inverse times the Higgs field. And these are all compatible with ADA. Oh and ADA is also compatible with the caramel ring in the sense that ADA of A and BC equals ADA of A, B and C. So a lot of structure. I'm not gonna tell you in general how to produce this structure although it's a super interesting subject. I put a few references. But I'm just gonna tell you what the structure is in the two examples that we're talking about. In the Sokel. Okay because I don't know what the Sokel is. You want me to write a formula? Okay. ADA of F and G. It's a cool formula. So ADA of F and G is the residue at infinity of F of X, G of X divided by W prime of X. That's what it turns out to be. But let me just write concretely what it is in these particular examples. So in the cubic case, in the same trivialization I was in before, this connection, this family of connections, novel H bar is the trivial connection by trivial respect to that basis, I mean, minus H bar inverse zero, zero, one, zero. That was the Higgs field I wrote before. And ADA is just zero, one, one, zero. That might make you think the whole thing is kind of trivial. Oh, let's first say another word about this. So this family of flat connections, I'm gonna wanna talk about a little more later. So let me right away make a point that you've seen this before probably, even if you don't exactly recognize it in this form. Namely, so this is a flat connection in a rank two bundle, but this is one of the kind that you can easily convert into a second order ordinary differential equation for scalars, linear scalar differential equation. And the way you do that is just define a section psi to be of the form f of z and H bar f prime of z, function of z, the novel H bar of psi equals zero is just equivalent to saying that the function f obeys the area equation. So H bar squared d by dz squared plus z times f of z equals zero. So this is a nice little second order equation. It's called the Aries equation, the Aries equation. Okay, and in the Cordic case, this family of connections novel H bar looks like this. It's again, the trivial connection minus one half H bar inverse times zero zero one z two z one zero, z two z one d z one minus H bar inverse times zero zero z two one zero z one zero one zero d z two. And so if you look at this, it looks 100% trivial. You say, look, all you did was you took the trivial connection and you added the Higgs field that you wrote down before. Why am I making a big fuss about this? The reason I'm making a big fuss about this is that this is not right. I have to put in the top corner minus H bar. Without that, nothing works. And so there's something, there is some elaborate machine that generates these things. I just want to make the point that it's a somewhat non-trivial structure, even in this very simple world of just families of polynomials over C or C two. Okay, but if you do exactly this that has all the structures that I'm claiming there. Okay, now, yeah. What is it? You mean is there some sort of abstract? I mean, that's the thing, the construction of novel infinity is kind of complicated. Let me give you some buzzwords. Is that where the H bar comes from? Oh yeah, that's right, that's right, that's right. This H bar is contributing to novel infinity, exactly. I didn't want to write a whole matrix of all zeros and a one half, so I snuck it in here. Yeah, yeah, so buzzwords for this structure are Frobenius manifold, and a buzzword for the construction of this is cytos primitive form, if you want to look this up, or look in the references and notes. Anyway, for my purpose, I want to take this structure as a given, and I just want to analyze it. So, okay, if someone gives you a differential equation, you want to know what are the solutions of this differential equation. So we want to study the flat sections, and let's actually start with just this area equation, which already has somehow the essential feature that I want to get at. The explicit ones that I'm going to construct have to do with boundary states of the Landau-Ginsberg model. Yeah, so, okay, so what are the NABLA H-bar flat sections? Well, so let's start with the area equation. So I actually want to write the solutions, it's a little easier to write them in this scalar language. So I'll write the solutions F of z. So for the area equation, there's a kind of explicit formula. Well, one explicit formula would be to say that the solution is the area function. That's a true fact. But it's somehow not very useful. So let me write a more useful formula. I'll write F sub i of z. So i is gonna run from one to two. And what I'll do is I'll take a contour integral of the function exponential of w of x divided by H-bar. Over some contour without boundary. So I'll call that contour C i. And then by little manipulation, which I put as an exercise, you can show that any such integral gives you a solution of the area equation. Now, so then the problem is, of course, to come up with a contour. So you first say, all right, no problem, you know, I'll just pick a contour. You know, I'll pick a contour like this. Well, okay, that gives a solution, but the solution is zero because, you know, this is an entire function, right? So you won't get anything interesting that way. You need to use contours that go to infinity, but then if you use contours that go to infinity, you have to be a little careful because in some directions, this function is blowing up as you go to infinity. So what we need is to go to infinity along some direction where this, the real part of w over H-bar goes to minus infinity. So we need the real part of w over H-bar to go to minus infinity at the ends of C-I, at the asymptotic ends of C-I. So how do you make such a contour? Well, one very convenient way of making them is to use a so-called left-shed symbol. So I think this already came up in a much, much more non-trivial context in Pavel Putrov's lectures. This is like the most baby example of the same thing. Unless he talked exactly about this example, did he talk about this example? No, he talked about it just in terms of Simon's like infinite dimensional space and so on. Okay, so here we're doing it just in C. So what's a left-shed symbol? Well, what you do is you start from a critical point. So let's say call this x1 and this x2, and we'll take these to be the critical point. So w' of xi equals zero. You remember from last time there are two, here there are exactly two critical points. And now we take the kind of steepest descent path through that point. So let me just sketch what they would look like here. There'd be one going like this, one going like that, and one going like this. So what do I mean by steepest descent paths? So ci is a left-shed symbol through xi. That's the steepest descent path, which means it has along this path, the imaginary part of w over h bar stays constant, and the real part of w over h bar goes to minus infinity. If you started the critical point, there's a unique path in each direction where that happens. You just follow it to minus infinity this way and follow it to minus infinity that way. In that way, you get some contour on which this integral is guaranteed to converge. And it gives you a solution of the area equation. So here's using these two critical points, I've got two solutions, that's the basis of solutions for this second order equation. So that's great. It is another good property. If you wanna actually, if you wanna think about these solutions, they have one very useful property which is that you can easily understand, so this is not an exact formula, but you can easily extract interesting information from this formula. In particular, one thing you can extract is the asymptotic behavior. I should say this picture is when h bar is real. So this picture, of course, depends on the phase of h bar. This is the picture when h bar is real. Now, what we can extract from this representation is we can extract kind of asymptotic information about what happens when h bar goes to zero. When h bar goes to zero, this integral is gonna be dominated by the contribution very, very close to the critical point. And there's a standard method of steepest descent for getting the asymptotics. What you get is that fi of z goes like, I think I got it written correctly, so square root of pi h bar divided by w double prime of xi, that part's not so important. What's important is e to the w of xi over h bar as h bar goes to zero for h bar real and positive. Yeah, I should have said real and positive. Okay, so that's the good news. The good news is that these functions have nice uniform asymptotics. That's a good property for a basis of solutions they have. Okay, there is some compensating bad news. The compensating bad news is let's think for a minute about what's gonna happen when we analytically continue in the parameter z. So this was for one fixed value of z. I've drawn the critical values here. This is x one is like square root of z and x two is minus square root of z. Those are the critical values. Now, as we change the value of z, these critical points move and at some moment something discontinuous is gonna happen, right? So here, in fact, in this picture, you already see a kind of weird asymmetry. I've got a contour here in this sector. I've got a contour here in this sector. I don't have a contour here. Why not? Well, okay, it just happened that way. But now, as I vary the parameter z, at some moment, the topology of these contours is gonna suddenly change. So as we vary z, the contour is ci can jump. Not only can, but will jump. So I guess I'll draw the picture for a different value of z. So suppose that the critical points now are here and here, then this contour is basically the same. This contour c one, but the contour c two, the contour c two makes a big jump as I change the parameter z. The moment when it changes, if you think about it, is exactly when,